WO2020064761A1 - Method, computer program product and computer device for evaluating volumetric subdivision models - Google Patents

Abstract

The invention relates to a method for the computer-aided evaluation of a volumetric subdivision model, which is described by a control point network (10) of control points (12) connected by means of edges (14), wherein the method comprises the following steps: − receiving information regarding a location which is to be evaluated and is situated in the volume of the subdivision model; − determining whether the location to be evaluated is in a volume element (52) that has an irregular edge (15) that is an edge (15) that connects a control point (12) with a valency other than six, − and, if this is the case: − determining an orientation of the irregular edge (15); − evaluating the volumetric subdivision model at the location to be evaluated based on the determined orientation. The invention further relates to a computer program product for executing such a method, and to a computer device for executing the computer program product.

Description

 Method, computer program product and computer device for evaluating volumetric subdivision models

The invention relates to a method, a computer program product and a

Computer device for evaluating volumetric subdivision models.

Technical systems, assemblies, workpieces or general objects often have to be checked for their properties or their behavior. In addition to real tests, computer-aided simulation has established itself as a common means of doing this. A prerequisite for a meaningful simulation is that an object is modeled as accurately as possible.

Typically, in a so-called design phase,

software-based CAD applications [Computer-Aided Design] creates a virtual 3D model of an object. This model is then examined in a simulation phase to determine how e.g. acting forces, temperatures or other influences, for which numerical simulation approaches are usually used. The simulation phase can also be called CAE [Computer-Aided Engineering].

In order to speed up the development process, information regarding the (in particular geometric) properties of the product, as it is available at the end of the design phase, must be provided in such a way that it can be done using a simulation environment (e.g. in the form of a computer program product) without extensive reformatting, if possible , Conversions or additional

Adjustments can be simulated. In the usual CAD-CAE workflow, the transition between design and simulation usually means converting the continuous CAD representation into a CAE or simulation-compatible, discrete geometry (e.g. in the form of discrete finite elements). This conversion is also known as "discretization" or "meshing". On the one hand, this step is time-consuming and often involves manual intervention and error handling. On the other hand, CAD geometry information is irretrievably lost. A conversion in the opposite direction to, for example, automated conclusions Drawing the simulation results is usually also not possible with the current approaches.

 In simulation systems currently on the market, the above is used

Conversion done to create a simulation compatible model. The simulation is often carried out using the finite element method (FEM). In addition to linear finite elements, elements with a square or cubic order are sometimes used, which give comparable results with less

Achieve degrees of freedom (see e.g. Daniel Weber, Johannes Mueller-Roemer, Christian Altenhofen, Andre Stork and Dieter Fellner: Deformation Simulation using cubic finite elements and efficient p-multigrid methods, Computers & Graphics 53 (2015), 185-195). Scientific work on simulation on continuous forms of representation has so far mostly been limited to representations with trivial NURBS geometry and can be summarized under the term “iso-geometric analysis - IGA” (see, for example, Thomas JR Hughes, John A Cottrell and Yuri Bazilevs:

Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer methods in applied mechanics and engineering 194, 39 (2005), 4135-4195). Furthermore, there is scientific work on the simulation of thin-walled objects (so-called “shells”) directly on limit surfaces through continuous facet decomposition (see Fehmi Cirak, Michael J Scott, Erik K Antonsson, Michael Ortiz and Peter Schröder: Integrated modeling, finite-element analysis, and engineering design for thin-shell structures using subdivision, Computer-Aided Design 34, 2 (2002), 137-148 & A. Riffnaller-Schiefer, UH Augsdörfer and DW Fellner: Isogeometric Shell analysis with NURBS compatible subdivision surfaces Appl. Math. Comput. 272, Part 1 (2016), 139-147). Other methods use the cell division or subdivision on which the invention presented here is based for a discrete simulation using the finite element method (Christian Altenhofen, Felix Schuwirth, Andre Stork and Dieter Fellner:

 Volumetry subdivision for consistent implicit mesh generation Computers & Graphics 69, Supplement C (2017), 68 - 79).

A distinction must also be made between models that focus on a pure

Limit surface description, and models that also take volume fractions into account or use a volumetric representation. The invention particularly relates to the last variant. One advantage of subdivision or cell decomposition is the avoidance of loss of information between a CAD model and a model used for the simulation. Often the same object description (or the same discrete control point network) can be used as the basis for both models, for example if the CAD models are also generated as a volumetric subdivision model. With the previously known modeling and simulation approaches, however, this is not always possible. In the context of the present disclosure, checkpoint networks are also generally referred to as control networks.

In detail, the invention presented here is based on a subdivision, which can also be referred to as a subdivision and / or cell division. In the subdivision, an output volume model and / or an output checkpoint network is used in accordance with

predetermined rules further subdivided, in particular in such a way that the resulting control point network is successively further refined. In theory, a so-called limit volume can be achieved, which represents the greatest possible refinement. Since carrying out such a maximum limit refinement is computationally complex, a direct mathematical evaluation of the limit is preferred instead, ie a mathematical calculation and evaluation of points of the limit volume without actually having to form the limit volume completely. An example of an algorithm that can be used for this is the Catmull-Clark algorithm for volumes (Catmull Clark Solids), the rules of which are summarized, for example, in the following scientific work, which is also referred to below as “Burkhart et al.”: Daniel Burkhart, Bernd Hamann and Georg Umlauf: Iso-geometric Finite Element Analysis Based on Catmull-Clark: Subdivision Solids In Computer Graphics Forum, Vol. 29.Wiley Online Library, 1575-1584. Reference is also made to the following

Scientific work: The refinement rules for Catmull-Clark solids, Kl Joy, R

MacCracken, Citeseer, 1999.

The Catmull-Clark algorithm generally provides for a refined network to be calculated on the basis of an initial network of control points connected by edges, new control points being determined in each subdivision step, which subdivide the network into cells with a smaller area (or with a smaller volume in three dimensions). A problem with such subdivision models is often that they merely approximate the shape of an object volume (or even just the object surface) via a grid of control points. In particular, the individual control points can be defined as so-called nodes or control nodes by functions describing the object shape and in particular surface shape (for example by B-spline functions). Therefore, one would like to evaluate a specific location of the object, for example, on its surface or in its volume (i.e. examine this location in the context of a simulation and, for example, determine the displacements and / or forces or other properties occurring there and / or the limit position and Gradients of this point), one cannot therefore always fall back on one of the control points of the subdivision model. This problem is illustrated by way of example from the attached FIG. 1, in which for one

two-dimensional case, a cell of a subdivision model consisting of four control points 100 is shown. These describe a circular object shape 102 by means of the

Control points 100 of defined B-spline functions, but without being part of the object 102 itself or lying within its area.

This problem is dealt with in particular in the following scientific work: Jos Stam: Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values, In Proceedings of the 25th annual Conference on Computergraphics and

Interactive techniques ACM (1998), 395 ^ 104. The considerations and proposed solutions presented there are also summarized in DE 699 15 837 T2, which

contains essential basic ideas and calculation approaches on which the present disclosure is also based. The content of DE 699 15 837 T2 and also of its US equivalent US 6 389 154 B1 are therefore fully incorporated into the present disclosure by reference. However, one difference between these teachings and the present disclosure is that there is only one

Surface models are considered in three-dimensional space, whereas the present disclosure relates to three-dimensional i.e. volumetric cases where checkpoint networks with cells that are connected and define the volume are considered.

DE 699 15 837 T2 describes the problem of directly evaluating any location in the subdivision model in such a way that the coordinates of the control points of a subdivision model can define basic B-spline functions to describe the contour of individual patches within a two-dimensional (patch) model of an object. The checkpoints can be general

Space coordinates can be defined (for example, as X, Y and Z values). Since the

However, as shown above, control points do not always match actual locations within the area, another parameter set is used that actually describes locations within the area or is defined in a corresponding coordinate system of the area. in the

In a two-dimensional case, a so-called u-v coordinate system or a u-v parameter set is defined for this.

Using the B-Spline basic functions, which describe the area itself, the u-v parameters can be translated into three-dimensional spatial coordinates in space. In other words, the corresponding X, Y and Z values can be determined for a location defined in the area by means of u-v parameters using the basic functions. Generally speaking, a relationship between the parameters for the locations in space and the area definition is defined by the B-spline basic functions in order to finally be able to specify the coordinates of any object location in a preferably ordinary Cartesian spatial coordinate system (for example in an ordinary X-, Y and Z coordinate system). As explained below, the basic functions can do more than just the object shape

Define object properties, such as a material density or a material type. Consequently, the described parameterization can also be used to determine specific properties of a given u-v location using the basic functions.

If each object location is mathematically clearly defined and / or its coordinates (preferably in the form of XYZ spatial coordinates) as well as gradient (derivation of the coordinate according to u and v for surfaces, or according to u, v, w for subdivision volumes) or properties using the B -Spline basic functions of the subdivision model can be determined, this model can be evaluated at any location. Reference is made to the description of the corresponding parameterization

The procedure described in DE 699 15 837 T2, for example in connection with FIG. 4 there. Evaluation can also be carried out within the scope of the present disclosure of a location in the subdivision model which comprise the above-described or parameterizations.

An equation which is explained in more detail below and which is used for evaluating any location p, which is a location on the limit surface and can therefore also be referred to as a limit location or limit point, with the parameters u, v can be found in two-dimensional case can be formulated as follows:

The matrix C contains control points of a surface element under consideration and the matrix contains the coefficients of the bicubic B-spline basic functions on the

corresponding checkpoints. In the three-dimensional case, this equation would be expanded by a further dimension w.

As further explained in detail in DE 699 15 837 T2, the evaluation at any location is only possible with little effort if the corresponding location is in a piece of land which is a so-called regular set of

Includes checkpoints. In particular, it is a piece of land that is spanned by a network of 4 x 4 control points and only regular

Control points contains (i.e. the control points within the network must be regular, whereas control points on the edges or outside edges of the network do not necessarily have to be regular). A checkpoint is said to be regular if it has a valence (or even a degree or node degree) of four. The valence corresponds to the number of edges that originate from or meet at a control point.

If, on the other hand, areas are considered which contain control points with one of four different valences (i.e. the control point with one of 4 different valences lies in the area and not on or on the edge of it and is therefore in the

Area), the coordinates of any location within this area cannot be determined by means of the parameterization explained above or the set of basic functions. Such checkpoints are commonly referred to as irregularly, extraordinarily or extraordinarily designated and can be due to the initial network, which was defined, for example, by an operator for the description of the object shape. In such output networks, it is not always possible to avoid irregular checkpoints from the outset for a precise description of the shape. DE 699 15 837 T2 therefore proposes approaches to enable the evaluation of arbitrary locations even within non-regular patches. As mentioned, these approaches are limited to the two-dimensional case.

Briefly summarized, it is recognized in DE 699 15 837 T2 that a network with an irregular checkpoint becomes a regular network by (arithmetically) performing a certain number of further subdivision steps. In other words, new control points are defined and / or calculated in the course of these subdivision steps, which ultimately lead to a piece of land (in particular a piece of land defined by a network (output network)) with an originally irregular control point being divided into pieces of land with only regular control points . The subdivision is carried out in such a way that the area or its output network is subdivided until the control point to be evaluated lies in a regular 4 x 4 control point network which is smaller than the output network. With each subdivision step, the areal portion of the checkpoint network containing the irregular checkpoint and resulting from the subdivision is reduced. More specifically, the subdivision of the irregular starting network or

Area in three regular and one irregular area, which, if it contains the irregular control point, could in turn be further subdivided, the area share of the irregular area in the total area of the original starting network decreasing with each subdivision step. However, in order to reduce the associated calculation effort and, in particular, to achieve a constant calculation time, DE 699 15 837 T2 calculates the intrinsic structure of an associated subdivision matrix in the area of an irregular control point.

This enables a mathematical determination of any number of local subdivision steps and related new control points. Since only that

Area that contains the irregular control point, is subdivided by calculation, but not necessarily the entire model, this can be called a local subdivision. For the described subdivision, some matrix expressions can be calculated in advance and for the arithmetic execution of the further subdivision steps can only be read from a memory. Thus, this calculation can be carried out essentially with a constant regardless of the position of a location to be examined and also the additional subdivision steps required

Calculation time to be executed.

As soon as the regular control points for a corresponding area have been calculated, any location within the area can be evaluated in a simulation using a set of B-spline basic functions. A detailed description of the calculation steps required for this can be found, for example, in paragraphs [0042-0088] of DE 699 15 837 T2, for example in paragraph [0080] dealing with information that can be calculated and stored in advance.

However, the principles set out in DE 699 15 837 T2 cannot be applied to volume-metric subdivision models. As a result, such volume-metric subdivision models have so far not been able to be used in the desired manner for simulating object properties, for example to calculate forces or stresses acting in the object and to suitably adapt the construction of the object. There is therefore no satisfactory way to take advantage of the real advantage of subdivision models in the form of the described slight loss of information during the transition between CAD and CAE in the context of corresponding simulations.

Thus, from the scientific work by Burkhart et al. a solution is known in which in volumetric models irregular volume elements can be subdivided locally so often until they become regular and can be evaluated using conventional calculation approaches. However, due to the required number of subdivisions to be calculated, the computing time is heavily dependent on the extent of irregularities and can generally assume undesirably high values.

The object of the present invention is therefore to better coordinate a computer-aided design phase and simulation phase in the development of technical objects or systems, in particular in such a way that precise determinations of relevant properties are made possible with a limited computing time. This object is achieved by a method, a computer program product and a computer device according to the attached independent claims.

Advantageous further developments are specified in the dependent claims. Furthermore, it goes without saying that the features mentioned in the introductory description can also be provided individually or in any combination in the solution disclosed here, unless otherwise stated or evident.

The invention solves the technical problem that computing times of a computer for simulations of the above type should not be arbitrarily high. The technical means proposed for this purpose is to align the irregular edges

take into account or make the basis for a simulation. As shown below, this enables extensive results to be saved beforehand (esp.

To use matrix expressions), which advantageously do not have to be recalculated and set up for each individual simulation. In other words, as a technical means it is possible to read out previously calculated results from a memory as necessary and to include them in a simulation calculation in order to limit the computing time of a simulation computer. An important concept of the invention, which enables the results stored in advance to be used and which also makes it possible to determine which of these results are to be read out and applied, is the consideration of the alignment of irregular edges.

The inventors have generally recognized that it is possible to use a volumetric

To evaluate the subdivision model in the knowledge of a spatial orientation of essential irregular properties by means of limited computational effort and possibly even an essentially constant calculation time regardless of the extent of the irregularity and / or further subdivision steps that may be required. In particular, it was recognized that by determining the alignment of a so-called irregular edge, which lies in a volume element of the subdivision model containing the location to be evaluated, the corresponding volume element can be divided into regions and / or planes that can be evaluated regularly and irregularly. In particular, an analog evaluation can be used in planes which are essentially orthogonal to the irregular edge, as is known in the prior art for the two-dimensional case (see for example DE 699 15 837 T2). These levels can also be based on the determination of the orientation of the irregular proposed according to the invention

Property or edge can be determined. On the other hand, in planes that contain the irregular property or edge, a regular evaluation according to the two-dimensional case can be carried out. To an actual one

To carry out three-dimensional evaluation, it can further be provided to combine the irregular but evaluable levels according to the two-dimensional approaches with basic functions along or in the direction of the irregular property. This is also made possible by first determining the alignment of the irregular property or edge. In particular, in this way irregular structures can be converted into so-called layered structures, which are explained below and which are regular in at least one dimension and are irregular in planes or layers extending to this dimension. These can then be in constant time (i.e.

constant calculation time) are evaluated, regardless of the degree of irregularity.

More precisely, the previous procedure in the two-dimensional case is summarized using equations (9) to (11) of this disclosure, all of which

Expressions between "C" and "N" depending on the n-subdivisions required for local evaluation can be saved in advance. Based on this, those also described in the referenced work by Jos Stam

Eigen basis functions cp (u, v) from equation (1 1) are formulated and largely pre-calculated.

As shown, for the three-dimensional case e.g. in Burkhart et al. so far taken a different path. To compensate for local irregularities, there was also local (computational) subdivision, but without short and, above all, constant computing times being achieved.

A surprising finding of the invention is in particular that irregular elements of a volumetric subdivision model taking into account the

Alignment of irregular edges against Burkhart et al. with a reasonable

Computational effort can be evaluated. In particular, it was recognized that, as in the 2D case, this enables comprehensive matrix expressions to be calculated and stored beforehand, which limits the computing time. For example, it was recognized that so-called layered structures can then be defined, after which irregular 2D planes of the same type along a third

Dimension can be strung together or layered. The third dimension preferably extends along the irregular edge, the orientation of which is determined and taken into account as an essential new feature according to the invention. A coordinate system used for evaluation can therefore preferably be aligned along this edge with one of its dimensions.

According to equation (17) below, this makes it possible to reuse the self-basis functions cp (u, v) known from equation (11) of the 2D case and for the most part previously calculated, and with a further basis function, which is also aligned along the irregular edge, to multiply. This is discussed below by way of example using equations (17) - (19) and, inter alia, FIG. 7b and FIG. 9.

The invention is characterized in particular by the following articles and findings, each of which is explained in detail below and which are partly purely optional:

 - Direct evaluation of layered structures with an irregular edge, for which the 2D basic functions of Stam can be multiplied with regular basic functions in the third dimension corresponding to an orientation of the edge and the evaluation takes place in constant computing time;

 - Building on this: Direct evaluation of marginal cells of a subdivision model, for which the so-called increase basic functions can be used in one dimension perpendicular to the edge of the subdivision model. This can be used for regular and stratified marginal cells and can also be done in constant computing time;

 - further building on it: a constant and / or consistent

Numbering method for the evaluation of completely irregular cells that have more than one irregular edge. The control points are preferably numbered according to a fixed scheme and are consistent for the particular irregular configuration found in the cell. This consistent numbering allows a calculation of the own structure of the subdivision matrix and the transfer of Stams evaluation algorithm to such irregular ones volumetric subdivision models, which is why irregular cells with more than one irregular edge can be evaluated in constant time.

The invention can advantageously be used for technical purposes as follows:

Finite element analysis (FEA) and / or isogeometric analysis (IGA), e.g. for simulating heat propagation or for structural mechanics. For

 Calculation of the simulation results or to solve the simulation scenario must be calculated at different points in the simulation algorithm integrals via finite elements (e.g. to calculate a stiffness matrix or a volume of the component to be simulated). If volumetric subdivision models are used as the basis for the simulation, as generally provided in the present case, the procedure according to the invention can accelerate the calculation of these integrals and thus the calculation of the entire simulation, since several locations (ie integration locations) have to be evaluated for each cell (or each element) . Such an evaluation of any location succeeds

 according to the invention by using previously stored expressions / results with low and even constant computing time.

Additive manufacturing processes, especially 3D printing with multiple materials, e.g. using the PolyJet process. In order to manufacture a component additively, it is broken down into layers for most 3D printing processes (so-called slicing). With 3D printing processes that can process several materials in one component, the material value to be printed is required for each location within each slice. With volumetric subdivision models, as commonly used herein, multi-material 3D printing models can be efficiently represented and even graded

 Model material transitions. Both to determine the slices and to determine the material value at each location in the slice, (sometimes multiple) evaluations at any location and, more precisely, limit locations are one

Subdivision model required. With the solution disclosed here, this succeeds much faster than with previous methods and even with constant computing time per location to be evaluated. In detail, a method for computer-assisted evaluation of a volumetric subdivision model is described (or, in other words, defined) by a checkpoint network of control points connected by edges, the method comprising:

 - Obtaining information regarding one in the volume of the

 Subdivision model lying to be evaluated;

 Determining whether the location to be evaluated lies in a volume element that has an irregular edge, which is an edge that connects a control point with one of six different valences,

 - and if this is the case:

 - determining an alignment of the irregular edge;

 - Evaluation of the volumetric subdivision model at the location to be evaluated based on the determined orientation.

All of the measures listed can be carried out in a computer-assisted manner, for example by means of a computing or processor unit of a conventional PC, for example a desktop or laptop, a server, a handheld device, for example a smartphone, and / or the computer device disclosed herein. In particular, a combination of such computers (for example a computer network) can also be used for execution. In particular, the measures according to at least one of the last four structuring points of the preceding list can be carried out by such a computer or a combination of such computers.

Intermediate and / or final results can be stored in at least one storage device of the computer or the computer network. In general, the method can include a step of storing the evaluated location in a

Storage means or, in other words, in a computer memory.

In a manner known per se, one control point can have several neighboring ones

Control points must be connected via individual edges, ie there are no further control points on the edges and at the ends of the edges. A plurality of control points connected to one another by means of edges can define a so-called (virtual) face in the subdivision model. More precisely, an area can be delimited and / or spanned by a plurality of control points, one of the control points being connected to two others from the plurality of control points by means of edges. If this area lies in the area of the surface of the subdivision model, it can accordingly be one

Surface area or, in other words, a partial surface of the surface of the

Act subdivision model.

The subdivision model can generally be used to describe and / or model an object, a workpiece, an assembly, a technical system or the like. In contrast to the two-dimensional case, a volumetric subdivision model can generally define and / or include a three-dimensional volume and also model the interior of the volume using the cells explained above. In particular, properties in a

Depth dimension (i.e. in depth and / or inside the volume) and not only in one plane on the surface can be determined and / or modeled.

The control points can be defined in the manner described above as control points or, in other words, control points or control nodes of functions which describe properties of the subdivision model. In particular, such functions can describe a form of the subdivision model, for example its so-called limit volume. The latter corresponds to the continuous volumetric body, which is generated by conceptually infinite refinements of the checkpoint network structure of the subdivision model and can be mathematically determined for many subdivision methods. A comparatively high or the maximum possible number of subdivisions of the subdivision model results in an approximation to this limit volume.

Alternatively or additionally, such functions can also generally

Define material properties, material transitions, material densities or the like, which can be done individually for each node. This is advantageous, for example, in order to describe multi-material workpieces or assemblies, for example by means of Rapid prototyping processes (or general generative layer construction processes) can be produced.

A location in the volume of the subdivision model to be evaluated can also be understood to mean locations in or on a surface of the volume or of the subdivision model.

In addition to the edges, control points and surfaces already described, the subdivision model can also include volume elements and / or cells and / or be defined by means of them. A volume element (English: cell or element) can generally be understood to mean any volume portion of the subdivision model including the edges, control points, etc. contained therein. In particular, it can be a volume fraction that can be evaluated using the regular calculation approaches explained below or that can generally be evaluated using preferred evaluation options. An example is a volume element that is defined or spanned by 4 x 4 x 4 control points and that can include 3 x 3 x 3 cells. For a

Evaluating can generally identify those 4 x 4 x 4 checkpoint arrangements that contain a location to be evaluated. In particular, the volume element can be a local (control point) neighborhood of a location to be evaluated and, for example, a cell which contains the location to be evaluated. More precisely, the location to be evaluated lies in a specific cell, with this cell and its neighbors and the associated control points being considered for the evaluation of the location. The cell containing the location to be evaluated can accordingly be arranged centrally or centrally (for example centrally in a regular 4 x 4 x 4 control point network or also centrally in an initially irregular one

Checkpoint network). A volume element can include at least one cell.

A cell can in particular be defined and / or surrounded by eight surfaces which are opposite one another in pairs and preferably parallel to one another in pairs. Generally speaking, a cell can be defined by any number of areas that completely surround and / or enclose a volume within the subdivision model. The corresponding volume can then form the cell. It it goes without saying that the cell is also delimited by corresponding control points and edges of the areas or comprises them.

The information regarding the location to be evaluated can relate to coordinates or parameter values of the location. In particular, it can be

Act coordinate values that indicate a position of the location in the model volume, for example in a spatial coordinate system, which can also be referred to as an X, Y, Z coordinate system. Alternatively, a cell containing the location to be evaluated can be identified and parameter values can be specified which define the position of the location in a local coordinate system of this cell. This local coordinate system can, for example, be referred to as a u, v, w coordinate system.

The subdivision model can be described and / or stored using known data formats or general computer-aided formats. Properties of the subdivision model sought can be determined from the corresponding data sets using conventional algorithms. For example, the valence of control points and consequently also the presence and / or a position of an irregular edge can be determined. Extentions, positions or orientations of individual volume elements or cells can also be determined or defined, and in particular those cells which contain the location to be evaluated. In principle, however, it is also possible to represent the subdivision model virtually and to enable an operator to manually identify irregular properties such as control points or edges and / or individual volume elements or cells within the

Select subdivision model manually.

According to the present method, in particular volume elements which have or comprise irregular properties (or also cells which are comprised by such volume elements) can be determined, for example on the basis of

corresponding records. An irregular edge can also be defined to have one of four different valences, which valence can be defined as the number of faces adjacent to the edge or, in other words, the number of faces originating from the edge and / or are connected via or with this. The evaluation can be understood to mean the parameterization described above, in particular the determination of location coordinates or properties of or at the location to be evaluated using functions which are defined by the control points. These functions can generally be basic functions or, in the case of regular cells, (in particular trivial) B-spline functions (modified B-spline functions can be used instead for irregular cells). The alignment of the edge can be used to identify areas (e.g. planes) in a volume element under consideration that can be evaluated using preferred and / or conventional approaches (e.g. using the two-dimensional approaches from the scientific work of Jos Stam or from the DE 699 15 837 T2). These can generally be irregular or regular areas or levels, wherein the irregularity can in turn depend on whether there are control points with predetermined valences and in the case of two-dimensional levels with one of four different valences.

It should also be pointed out that if the location to be evaluated is not in a volume element and / or a cell that has an irregular edge, a location evaluation can be carried out using regular approaches. In particular, the volume element can then be defined in the form of a local neighborhood around the location to be evaluated (or the cell which contains this location), for example in the form of a 4 x 4 x 4 control point network or an arrangement of 3 c 3 c 3 cells. One possibility for evaluation is then to determine trivariate basic functions from products of univariate basic functions of the control points of this volume element.

In general, the following sequence is possible within the scope of the present invention, in which not all measures are mandatory and the sequence of which is not restrictive:

It is determined whether the volume element with the location to be evaluated has a completely irregular structure, ie no dimension or plane can be defined, along or in which there are regular properties (ie regular edges and control points); and if this is the case (if not there is already a layered structure and the following step is skipped):

 A number of (arithmetic) subdivision steps are determined (e.g.

 with reference to the following FIG. 9) until a structure which is regular in at least one dimension is obtained (so-called layered structure, as will be explained in more detail below), entries for a subdivision matrix being selected in a predetermined manner for this subdivision, in particular depending on the structural properties of the Checkpoint network; and if the layered structure has been reached or already existed:

 As explained above, an alignment of the (still) remaining irregular edge is determined and then evaluated based on the location under consideration, in particular by means of subsequent (subdivision) basic functions and / or a calculated intrinsic structure of the subdivision matrix.

One embodiment also provides for the following measure:

 - Define a coordinate system according to the orientation of the

 irregular edge and in particular such that the irregular edge extends along a (e.g. predetermined and / or preferred) axis of the coordinate system.

Defining the coordinate system according to the orientation of the irregular edge means that the orientation of the irregular edge when defining the

Coordinate system is taken into account. This coordinate system can be the same coordinate system to which the information relating to the location to be evaluated also relates. In particular, it can be the coordinate system explained above for defining locations in a single cell, for example a so-called u, v, w coordinate system. Such cell-related coordinate systems can be distinguished above all by the fact that their axes, which preferably each run orthogonally to one another, each have values from 0 to 1 and the cell is parameterized according to these axes.

Alternatively or additionally, it can be the same coordinate system in which the control points are defined, or a different one

Coordinate system. Within the scope of the present disclosure, all

Coordinate systems can be defined as Cartesian coordinate systems. It goes without saying that the definition of the coordinate system can in turn be virtual or arithmetic and generally computer-aided and / or by manual input by the user.

In relation to a cell-related (u, v, w) coordinate system, this can

can be defined, for example, in such a way that the irregular edge extends, for example, along a corresponding u, v or w axis. In particular, in a first step, the alignment of the irregular edge in the coordinate system and, for example, also a relative alignment to at least one axis of the

Coordinate system can be determined. The coordinate system or the subdivision model can then be realigned comprehensively around the irregular edge, so that the irregular edge extends along the predetermined axis. For this purpose, for example, the coordinates of the control points and / or edges of the subdivision model can be transformed in a manner known per se in accordance with the adjustment to be carried out.

The inventors have recognized that conditions can be achieved in this way in which a volume element, which contains a cell with the location to be evaluated, can be divided at least after at least one further (computational) division into regular planes along the irregular edge and irregular planes orthogonally thereto. In particular, this enables a layer-by-layer structure and / or a layer-by-layer evaluation of the volume element to be achieved, in which two-dimensional node networks or layers of essentially or completely identical construction are strung together along the irregular edge. In particular, these layers can comprise irregular control points at the same positions in each case.

For such a layer-by-layer observation, volume elements are preferably used (ie defined and / or selected in the subdivision model), which are spanned in at least one dimension (preferably even two or three dimensions) by 4 nodes and / or 3 cells, this state in turn after at least one further (computational) subdivision can be achieved.

The following measures are also provided for further training: Identifying a cell of the subdivision model which contains the location to be evaluated; and

 Determine the number of divisions of this cell that are required for the location to be evaluated to be in an evaluable cell that results from the division from the original cell.

Optionally, it can also be provided that the subdivision (at least locally, that is, in the area of the cell) is carried out computationally in order to create an evaluable environment around the location to be evaluated. The arithmetic subdivision can take place with a (predetermined) constant calculation time.

In this context it can further be provided that the evaluable cell is characterized in that it does not comprise a control point with one of six different valences. This applies, for example, to the case of a regular cell described below. However, the teaching disclosed herein can also be used to evaluate cells that are present in a layered structure explained below or (at most) have an irregular edge. The number n of necessary subdivisions for a location to be evaluated at the position u, v, w in one

Cell-related coordinate system of a cell can generally be determined using the following formula: n = mm j-log _^ , <H). - log-, (t>). - log-, (w) | ]

Additionally or alternatively, the number of subdivisions required can be determined by means of the relationships explained below with reference to FIG. 9, which in turn are based on a cell-related coordinate system and in particular a predetermined alignment of the irregular edge therein.

The cell can be divided into eight smaller ones in a single step

Individual cells are subdivided, which, when put together, occupy the same volume as the output row. If the location to be evaluated is then within a (new) cell that no longer includes the irregular control point, it can be evaluated directly by means of approaches shown in the exemplary embodiments. However, if the (new) cell containing the location to be evaluated still has the irregular checkpoint, at least this cell must go further be divided. The relationship between the number of required

Subdivision steps depending on a distance of the location to be evaluated from the irregular control point can be determined beforehand and stored (see also illustration of the corresponding relationships in the following FIG. 9), so that depending on the specific parameters of the location to be evaluated, a computational local division of the subdivision model is carried out immediately can be.

The evaluation of the cell which contains the location to be evaluated can be carried out using special subdivision basis functions, as have already been explained above for the two-dimensional case. This offers the advantage that at least some of the previously calculated matrix expressions and / or matrix-matrix multiplication can be used and the required computing time can thus be reduced and in particular kept essentially constant.

By determining the required number of subdivisions, the given location can be evaluated with little effort, in particular with knowledge of the alignment of the irregular edge and preferably by means of partially already pre-calculated matrix expressions and / or matrix-matrix multiplication (for example as part of the subdivision basic functions).

Overriding can also be provided that trivariate basic functions are used for the evaluation at the given location or the given location, which are based on basic functions along an axis of the coordinate system that runs along the irregular edge.

Furthermore, it can be provided in this connection that the basic functions for the evaluation are also based on bivariate subdivision basic functions that are determined in planes spanned by the remaining axes of the coordinate system.

An equation from which the above-described calculations emerge is reproduced below and is explained in more detail in the context of the exemplary embodiments. The bivariate subdivision basis functions (p (u, v) relate to one in the Scientific work by Jos Stam or term defined in DE 699 15 837 T2 for evaluating irregular two-dimensional patches (referred to there as an own basis function, but referred to herein as subdivision basis functions due to the necessary adjustments for the three-dimensional case explained below). On the basis of the determination of the alignment of the irregular edge provided according to the invention and the axis definition along this edge provided in the context of the further developments described, this term can also be used to multiply it by basic functions N, (w) along the axis (w) along which the irregular edge also runs. The resulting trivial basic functions that can be used in the form of the equation (15) explained below to evaluate a location p (u, v, w) are:

The following equations (18) - (19) can also be used for evaluation.

Here, j denotes the respective control points in an i-th u, v plane or layer, which are each orthogonal to the irregular edge along the w axis. In addition to the basic provision of a practical evaluation option for the three-dimensional case, one advantage of this option is that the calculation effort is significantly reduced by using the subdivision basis functions (p (u, v). In particular, an essentially constant calculation time can be achieved, regardless of The extent of any irregularities Furthermore, essential matrix expressions on which the subdivision basis functions (p (u, v) are based can be partially precalculated and read out with little effort for the evaluation.

In general, it can first be checked whether the location to be evaluated is in a so-called layered structure. If this is not the case, the layered structure can be created as part of a preparatory computational subdivision. For this purpose, a subdivision matrix can be defined in a predetermined manner in order to achieve subdivision in a constant calculation time. In a further development, if the volume element does not have a regular structure in at least one dimension, the volume element is subdivided and evaluated by means of a subdivision matrix (see e.g. the following example for subdivision matrix A) and the entries in the subdivision matrix depend on at least one structural property of the checkpoint network. Both

Entries can generally be the subdivision-based weighting factors for checkpoints or checkpoint coordinates. These should be entered in the subdivision matrix (i.e. with preferred column and row positions) in order to enable evaluation in a constant calculation time. An example of this is described below, in which numbering or indexing and, based on this, entry in the subdivision matrix takes place in a specific sequence according to three different groups of control points. In particular, so-called changes of direction can be prevented during successive subdivisions, which would increase the calculation time linearly with the required subdivision steps.

The structural property can be a position of a checkpoint with one of six different valences (also referred to herein as irregular checkpoint) within the checkpoint network. More specifically, an irregular checkpoint can be determined and, for example, according to a distance from this irregular checkpoint

Checkpoint numbering or indexing of the other checkpoints is carried out in order to enter them in the subdivision matrix.

Furthermore, if the cell with the location to be evaluated is bounded by a single irregular edge, a control point which is an end point of the irregular edge can be selected as the irregular control point for specifying the structural properties and the entries in the division matrix.

Alternatively, if the cell with the location to be evaluated is delimited by a plurality of irregular edges, a control point which is connected to all irregular edges can be selected as the control point for determining the structural properties and the entries in the division matrix. However, if there is no control point connected to all irregular edges, the Checkpoint network can be divided once to achieve the above-mentioned state.

According to a further development, the volumetric subdivision model is also described in

Evaluated depending on whether the location to be evaluated is in a border area of the subdivision model to the environment. The inventors have generally recognized that in such a state, further simplifications of the evaluation are possible and / or special evaluations are required. A boundary region can generally be understood to mean a region and / or volume element of the subdivision model that is positioned near an interface of the model to the surroundings and / or that contains such an interface. In particular, the border region can be a neighborhood of at least 3 c 3 c 3 control points or at least 3 x 2 x 2 cells around the irregular property (for example the irregular edge)

Include property contained cell.

As shown below, it is possible due to special features recognized by the inventors to use fewer control points or cells for evaluation than for locations and / or cells located in the volume of the subdivision model. In particular, possibilities were recognized for using so-called sharp-edge functions known from the prior art in corresponding border areas, as a result of which the number of required control points and cells and the general computing effort are reduced.

For example, if the location to be evaluated is located in a border area of the subdivision model to the surroundings, at least one basic function used for the evaluation can be determined based on a sharp edge function along the axis along which the irregular edge also runs. This can be seen, for example, from equation (3), which is explained in more detail below and below, and which can be used in equation (15) below

Basic functions B are considered for a solid element considered at individual nodes i. In the directions that are orthogonal to the irregular edge (in the example below the u and v direction), conventional basis functions are considered, while in the direction along the irregular edge (in the example below w) and in Direction of the edge or the boundary, a sharp edge function C, which is explained in more detail in the context of the exemplary embodiments, is considered. The square brackets are a round-up operator that rounds up the content of the brackets in the case of fractions to the next higher integer, whereas the% sign indicates a modulo operator:

Ni {u, v, w) = _{] 6} («)% 4l% 4 (v) N (w)

The evaluation described above and explained on the basis of equation (3) can take place in particular if it is determined that the irregular edge is orthogonal to or in an interface of the subdivision model with the surroundings. In particular, there should be regular structures in the u and v directions, and if this is not the case, the equation (19) explained below can be used.

In a further embodiment, if the location to be evaluated is located in a border area of the subdivision model to the surroundings and the location to be evaluated lies in a cell of the subdivision model which comprises more than one interface to the surroundings, the cell is evaluated at least once (computationally) before the evaluation. be divided. The inventors have recognized that, at the latest after such a division, it is possible to use one of the evaluation options explained above for border areas, for example on the basis of sharp edge functions.

The invention further relates to a computer program product comprising

Program instructions for executing a method according to one of the preceding aspects when executing these program instructions on a processor unit. The computer program product, in particular a computer program, is therefore designed to carry out the method according to the invention in one of the embodiments which are described in this description. In other words, the method can be carried out using software. However, it is not excluded that at least parts of the method can alternatively be carried out by a combination of software and hardware designed to carry out the method, in particular with at least one integrated circuit (such as an FPGA) designed to carry out the method, or exclusively by such hardware . Furthermore, the invention relates to a computer device comprising a processor unit and a memory unit in which a computer program product according to the

previous aspect for export by the processor is stored (in particular stored and / or loaded executable).

The invention is explained in more detail below with reference to the accompanying drawings. Identical features or features that match their type and / or function can be provided with the same reference symbols in all figures.

Fig. 1 is a schematic diagram for explaining a relationship of

 Control points and an object shape defined thereby;

2a-c examples of regular and irregular two-dimensional surface pieces;

Fig. 3 is an illustration for explaining the necessary number of

 Subdivision steps to obtain regular two-dimensional checkpoint networks;

4 shows a diagram for explaining the indexing of individual control points before and after an additional subdivision;

5a-b are a diagram for explaining sharp edge functions;

Fig. 6 is an illustration of a regular volume element of a volumetric

 Subdivision model in the form of a 4 c 4 c 4 network;

7a-b show an example of an irregular volume element of a volumetric

 Subdivision model;

8a-b another example of an irregular volume element

 volumetric subdivision model;

9 is an illustration for explaining the necessary number of

Subdivision steps to get regular solid elements; 10a-b show an illustration of the irregular volume elements from FIG. 7b and FIG. 8b after a further local subdivision;

Fig. 1 1 a-e examples for cells of a volume element that contain a location to be evaluated and which are at the interfaces of a control point network with the environment;

12 shows illustrations for explaining a required computing time; and

13a-e representations for explaining a numbering of control points.

In the following, some basic considerations of DE 699 15 837 T2 are first discussed, on which the solution disclosed herein is based in part. A complete explanation of the corresponding calculation steps can be found in the scientific work by Jos Stam mentioned above, but also in

DE 699 15 837 T2 with essentially the same content, in particular in paragraphs [0042- 0088]. The discussion of these previously known approaches relates exclusively to the two-dimensional case, whereas the solution presented here enables expansion to three-dimensional cases.

FIGS. 2a-c show examples of regular and irregular checkpoint networks 10, 11 for modeling two-dimensional surface pieces, such as those from one

two-dimensional subdivision model can be included. The solid lines and points show an output network 10 and the dashed lines and points show a network 11 after performing a subdivision step in accordance with the Catmull-Clark algorithm (also referred to below as subdivision network 11). It goes without saying that the subdivision model can comprise a plurality of surface pieces and thus also a number of corresponding control point networks 10, 11, which then

would be positioned next to each other or merge.

FIG. 2b shows a regular checkpoint network 10 that of 4 × 4

Control points 12 is clamped, which are connected to one another by means of edges 14. A plurality of edges 14 and control points 12 each close two-dimensional cells 13 between them. For reasons of clarity, not all control points 12 and edges 14 are provided with a corresponding reference symbol in all of the figures below. The control point network 10 serves to model or mathematically describe the shape of a two-dimensional surface piece, which is not shown separately.

Each point within the control point network 10 (i.e. with the exception of its edge) has a valence of four and is therefore regular. A 4 c 4 control point network 10, which contains only regular control points 12 (i.e. with the exception of the edge), defines a so-called regular bicubic B-spline surface piece or can be described mathematically as such. Coordinates and properties of locations within such a piece of land can be determined with little effort using the B-spline basic functions described at the beginning. An evaluation can therefore be carried out at any location within such a regular area.

Although this is not necessary in the case of FIG. 2b, the control point network 11 is also shown in dashed lines after a further subdivision in the form of a subdivision network 11 for reasons of clarity. It can be seen that this has a finer subdivision, since its control points 12 are much closer together. The fact that the subdivision network 11 and the output network 10 have common points can be attributed to the fact that the output network comprises control points 12 which are equidistant from one another and results generally from the

Subdivision rules.

FIG. 2a shows a control point network 10, again shown with solid lines, for modeling a two-dimensional surface piece, which is not shown separately. The network 10 contains an irregular or extraordinary point 18, which has a valence of three. The checkpoint network 10 is therefore irregular and cannot be evaluated easily using basic B-spline functions. In particular, no regular 4x4 checkpoint network 10 can be defined that contains only regular checkpoints 12 in its interior (partially overlaid by dashed checkpoints 12 of a subdivided checkpoint network explained below). Instead, it would only be possible to define a 3 c 3 network of control points 12 which are correspondingly provided with a reference symbol and which are the irregular ones Item 18 includes. However, this network does not define a bicubic B-spline patch or cannot be mathematically described and evaluated as such.

To remedy this, a further local subdivision is carried out according to the rules of the Catmull-Clark algorithm, which results in the subdivision network 11 shown in dashed lines, which in each case partially overlaps control points 12 of the output network in FIGS. 2a-c. As a result of the subdivision, three different regular 4 × 4 checkpoint networks can be defined, one of which, which also includes the irregular point 18, is outlined in a wave shape in FIG. 2a. This network is regular because it contains only checkpoints 12 with a valence of four inside.

 Evaluations can thus be carried out at all locations in the regular network 20 (i.e. also at positions between the individual control points 12), since their coordinates can be determined by means of conventional B-spline basic functions. If the location to be evaluated coincides with an irregular control point 12, theoretically infinite subdivisions have to be made, or it can be evaluated directly using n = infinity using the equation (7) explained below. In the latter case, so-called limit rules can be predefined, which say, for example, that if the eigenvalue matrix is raised to power, an entry specified with 1 retains its value, while all other entries between 0 and 1 assume the value 0. In general terms, only the largest eigenvalue of the eigenvalue matrix can be retained and all other entries can be specified with 0.

2c shows a further example in which a control point network 10 contains an irregular point 18 which has one of four different valences, namely a valence of five. The output network 10 shown again by means of solid lines (but also no possible partial network included hereof) thus does not define a regular 4 × 4 checkpoint network 10 which contains only regular points. To still

To enable evaluations of locations within the network 10, a further subdivision is made in accordance with the subdivision network 11 shown in dashed lines.

As a result, three different regular 4 × 4 networks 20 can be defined, one of which is indicated in a wavy outline in FIG. 2c. The further regular networks 20 would be obtained if the wave-shaped section in FIG. 2c were shifted upwards by one row or one column to the right. Within this regular Networks 20 can thus be evaluated again at any location and thus also between the control points 12.

Depending on the position of an irregular checkpoint 18 in an irregular starting checkpoint network 10 and in particular a relative distance of the location to be evaluated from the irregular checkpoint 18, there is a different number of

Subdivision steps necessary to ensure that the location to be evaluated is in a regular network or subnet. In general, the smaller the distance between the

location to be evaluated from the irregular point, the larger the number of division steps required.

This is also clear from the illustration in FIG. 3. The parameter space is generally shown in the form of a cell-related u-v coordinate system 17 of a two-dimensional cell 13 or a two-dimensional subnetwork that encloses such a cell 13 (cf. FIG. 2a). Each two-dimensional cell 13 can be described by means of such a two-dimensional cell-related u-v coordinate system 17, where u and v each take values between 0 and 1 (i.e. the cell 13 is parameterized in a corresponding value range).

If there is an irregular point in cell 13, the u, v coordinates (0,

0) assigned. Depending on the position (i.e. at which u, v coordinates) a location to be evaluated is located, the number of necessary subdivision steps can be read from this representation. More precisely, n denotes the number of subdivision steps to be carried out in succession and k the number of steps

Subdivision step each obtained regular cells or networks. Each

Subdivision step n three regular cells or networks k = 1, 2, 3 are obtained (hereinafter also referred to as regular patches) and a cell which still contains the irregular point 18.

3 shows a first location to be evaluated at a position 22 with the u, v coordinates (0.25; 0.75). Since this lies in one of the subdivision networks with the value n = 1, only a single subdivision step is required in order to obtain a regular cell (k = 3) comprising the location 22, which can then be evaluated. As a further example, a second location to be evaluated is shown at a position 24 with the approximate u, v coordinates (0.2; 0.1). Since this lies in one of the subdivision networks with the value n = 3, three successive subdivision steps are required in order to obtain a regular cell (k = 1) comprising the location 24, which then

can be evaluated.

This clarifies: The closer the location to be evaluated to the irregular point 18, the higher the required number of subdivision steps n to be carried out. Further background information on this representation and in particular a mathematical description of the subdivisions of the so-called two-dimensional shown in FIG

Standard squares can be found in DE 699 15 837 T2 in connection with FIG. 10 there (see in particular [0056-0060]).

An essential finding of Jos Stam's scientific work and DE 699 15 837 T2, which is essentially the same in content, lies in the fact that such necessary subdivision steps include by means of matrix multiplications, whereby essential matrix expressions can be partially saved in advance (i.e. before

Carrying out actual evaluations and / or simulations). Regardless of how many subdivision steps are required or would theoretically be necessary for the evaluation of a specific location, any locations within the checkpoint networks 10 or cells 13, which are in themselves irregular, can be evaluated based on this with a substantially constant calculation time by reading out and using the previously stored matrix expressions .

The calculations used, on which the solution disclosed herein is based in part, are briefly summarized below. More detailed information and

The background to this can be found in the scientific work by Jos Stam (see in particular Chapter 4) and DE 699 15 837 T2 (see in particular [0060-0087]).

First, reference is made to FIG. 4 and the illustration there to explain the indexing of individual control points 12 before and after an additional subdivision. Again one can see one shown with solid lines

Exit checkpoint network 10 containing an irregular point 18. The exit control point network 10 further comprises a plurality of control points 12 (not all with one corresponding reference numerals), which are denoted by 1 to 2N + 8. N denotes the valence of the irregular point 18. Furthermore, a subdivision network 11 is shown in dashed lines which, as a result of the subdivision, contains control points 12 which are numbered 2N + 9 to 2N + 17. Because of their distance from the irregular point 18, these are no longer required for further local subdivision and are therefore shown in dashed lines.

The starting point is a matrix Co, which contains the checkpoints 12 of the starting checkpoint network 10. More precisely, the location vectors of the individual control points 12 defined in an X, Y, Z coordinate system are combined as (K x 3) - matrix Co, where K is the number 2N + 8 of control points 12 that define the output control point network 10 . The numbering of the

Control points 12 are also used to define the content of the matrix Co. More precisely, this defines the line number at which the coordinates of a respective point 12 are sorted into the matrix Co. The x, y, z coordinates each result in a column value (i.e. the first column contains the x, the second the y and the third z coordinate).

With a so-called subdivision matrix (or also subdivision matrix), which is set up in accordance with the general Catmull-Clark subdivision rules, a matrix Ci can be determined, the 2N + 17 location vectors of the individual control points 12 of the

Subdivision network 1 1 contains. More precisely, the connection applies

Ci = ÄCo (4).

In order to carry out a plurality of successive subdivisions, only the upper K = 2N + 8 rows of the matrix have to be extracted, which results in a (K x K) matrix which is referred to below as A (see also [0051-0052 ] of DE 699 15 837 T2). This matrix can be used for any number n of

Subdivision steps are calculated in advance and stored (i.e. before the actual evaluation and / or an actual simulation). For any number of

The following relationship (5) therefore applies to subdivisions, where C _{n is} a matrix containing the location vectors of the control points 12 after n subdivisions of the initial Checkpoint matrix Co using the subdivision (or subdivision matrix) A represents:

 (5).

 The calculation of the intrinsic structure of A can greatly reduce the calculation effort for evaluating any location within the surface model, since n matrix-multiplications therefore do not have to be carried out for n subdivisions to be carried out (see also the following equation (7)). The following applies to a matrix V, which contains the eigenvectors of A as column vectors and L, which is a diagonal matrix with the eigenvalues of A along the diagonals:

AV = VA A = VAV ¹ (6).

If the diagonal form of A is inserted into equation (5) above, the following relationship is obtained, which only requires a limited number of matrix multiplications and exponentiation of a diagonal matrix, the exponentiation of the diagonal matrix against the matrix multiplications in particular that Equation (5) requires significantly less computing power and computing time:

 (7).

After the required number of n subdivisions has been made, the resulting regular network 20 can be evaluated on the basis of the control points 12 defining the network 20 by means of a bicubic B-spline area piece evaluation. These points, which are outlined in a wave shape for example in FIG. 2c for one of the regular networks 20, can be extracted from the matrix C _n by means of a so-called selection matrix P _k (also a recording matrix or English “picking matrix”), which is multiplied by C _n outputs the required points in the correct order

Background information for calculating the selection matrices P _k , with a respective associated selection matrix P _{K being able to be} defined for each area k = 1, 2, 3 from FIG. 3, is given, for example, in the scientific work by Jos Stam on page 19. The following relationship can therefore be defined for a location p to be evaluated with the coordinates u and v in the u, v parameter space from FIG. 3, the

Control points of the associated regular network 20 or regular area are multiplied with the B-spline basic functions N (u, v) at these control points:

Since the basic functions can specify and / or interpolate coordinates and / or properties at the control points (for example material properties such as density or temperature conductivity), equation (9) can be used to evaluate the location p an u, v in the sense of determining its coordinates and / or properties are made. The expression from equation (1 1) can also be referred to as a bivariate subdivision basis function.

As a result, the evaluation of any location in a possibly also irregular exit control point network 10 is made possible, for which purpose only the number of required subdivisions n has to be determined. This can be done, for example, using FIG. 3 and in particular a mathematical description or tabular summary of the relationships shown there (for example a table in which the required subdivisions n and optionally also a relevant area piece k are stored as a function of the u, v coordinates of a location to be evaluated are). Alternatively, the number of required subdivisions n can be determined using the following equation: n [min

A substantial part of equation (1 1) or (10) can be calculated in advance and stored in a memory. In the scientific work of Stam (see p. 1 1 -12) in particular the precalculation of the expressions V ^_T and VP for k = 1, 2, 3 is explained in detail. DE 699 15 837 T2 describes a computer-assisted implementation of the calculations described, in particular from paragraph [0080]. Accordingly, the potentiation of the diagonal matrix L as a function of the subdivision steps n is mainly or exclusively decisive for the actual calculation time, which, however, represents a comparatively low-cost arithmetic operation with an essentially constant arithmetic time. The total calculation time required can therefore be independent of the number n of local subdivision steps required

Turn out to be essentially constant, which, however, as already mentioned, is only possible with the previous approaches for two-dimensional cases.

An additional specialty for both two-dimensional and for

Three-dimensional subdivision models represent so-called sharp edges (also known as folds, English: crease edges). These can be defined by an operator, for example when creating an output network. They generally describe edges 14 within a control point network 10 at which no smoothing (i.e. finer subdivision) usually associated with advancing subdivisions or subdivisions is desired. Details on the definition of sharp edges and the special subdivision rules to be found there can be found in the following scientific work (see in particular pages 102-105 and 122 ff.): Sven Havemann: “Generative Mesh Modeling”, doctoral thesis 2005, available at https : //publikationsserver.tu-braunschweig.de/receive/dbbs_mods_00000008).

For sharp edges, the sharp basic functions explained in Havemann's scientific work are also considered ("crease basis functions"). These replace the usual basic functions from, for example, equation (9) when evaluating surface pieces which contain corresponding sharp edges. Sharp edges in volumetric models can generally be analogous to sharp edges in

two-dimensional models are dealt with, which is why a two-dimensional case is assumed as an example below.

A possible derivation of a sharp edge function is explained below and is based on the observations in Havemann's scientific work, according to which sharp edges have the same or equivalent effects as one Extrapolation of an edge at its end point. Reference is also made to the following articles: SCHWEITZER JE: Analysis and Application of Subdivision

Surfaces. PhD thesis, 1996; ZORIN D., KRISTJANSSON D .: Evaluation of piecewise smooth subdivision surfaces. The Visual Computer 18, 5-6 (2002), 299-315; KOSINKA J., SABIN M.A., DODGSON N.A .: Semi-sharp creases on subdivision curves and surfaces. In Computer Graphics Forum (2014), vol. 33, Wiley Online Library, pp. 217-226.

Referring to FIGS. 5a-b, a B-spline curve shown in dashed lines and defined by control points pi-p ₄ is shown in the left case, the control points pi-p _{3 being} collinearly and equally spaced from one another or also equally spaced from one another. The control point pi, on the other hand, represents the result of the extrapolation from p ₃ to p ₂ .

Overall, the following relationship can be established:

P1 - P2 = P2 - P ₃ (12).

The edge 14 extending from pi to p ₂ is an extrapolation of the edge 14 which extends from P ₃ to p ₂ . The corresponding B-spline curve or sharp-edge functions c can therefore be formulated as follows, with the B-spline base function (one per control point pi-p) in the final column vector:

 (13).

In this equation (13), if pi is replaced by 2 xp ₂ - p ₃ based on equation (12), the following is obtained:

(14). Using this function, the same curve segment from FIG. 5a with only three

Control points are defined as shown in Figure 5b. The cubic functions contained in the associated equation (14) represent the sharp-edge basic functions, the former assuming the value 1 for a value of u = 0 and decreasing for larger u values.

The following describes the new approaches proposed according to the invention

in order to be able to evaluate any location in volumetric subdivision models and not only in two-dimensional subdivision models or in their patches.

As in the two-dimensional case, volumetric subdivision models consist of control points, edges and areas delimited by them. They also include cells, each delimited by a set of areas. A volume fraction of the model in which an evaluation is to take place is also referred to below as

Volume element called. A special case is also represented by cells that lie in the outer edge areas of the model and in which at least some of the surfaces

Represent interfaces to the environment. These cells can also be called boundary cells.

Furthermore, in the volumetric case, extraordinary or irregular control points are defined as points that have one of six different valences. There are also extraordinary or irregular edges that have one of four different valences. In this case, the valence is defined as the number of surfaces adjoining the edge or, in other words, as the number of surfaces which start from and / or are connected by the edge.

A trivial case for evaluating any location in a volume metric

Subdivision model occurs when the location to be evaluated lies in a regular cell 52 without irregular control points and / or irregular edges. For such a cell 52, a neighborhood of the model can be considered in the form of a volume element 50 that is spanned by 4 x 4 x 4 control points 12 or 3 c 3 c 3 cells 52. Such a case is shown in FIG. 6, again not all of the above or elements mentioned below are provided with a corresponding reference symbol (the same also applies to the following further figures).

6 shows a central cell 52 which is delimited by six surfaces 54, each surface 54 being spanned by four control points 12 together with edges 14 connecting these control points 12. Each control point 12 of a surface 54 is connected to two of the other control points 12 of this surface 54 by means of edges 14. The cell to be evaluated in the example shown is located in this cell 52, which is why the corresponding neighborhood of 4 x 4 x 4 control points 12 or 3 x 3 x 3 cells 52 in the form of the volume element 50 is considered around this cell 30. It goes without saying that the subdivision model can consist of numerous further volume elements 50, which can adjoin the volume element 50 shown in FIG. 6 accordingly.

In this case, an evaluation can take place on the basis of trivial B-spline basic functions. Three dimensions or parameters u, v, w become one

cell-related coordinate system 57 considered that more specifically refers to the central cell 52 containing the location to be evaluated. The parameters u, v, w of this coordinate system 57 can each assume values from zero to one. On the basis of the 64 control points p of the volume element 50 shown, any location e in the central cell 52 can be evaluated as follows, taking into account the basic functions defined by the control points p:

The trivial basic functions N, (u, v, w) at the i-th control point are defined as a product of three univariate B-spline basic functions according to the following equation (16), which also corresponds to the aforementioned equation (3), where the square brackets again denote a round-up operator and the% sign a modulo operator:

The evaluation of a location in the cell 30 shown in FIG. 5 is thus possible without considerable computational effort and per se in a constant calculation time. Again, the basic functions can specify coordinates and / or properties of or at the corresponding control points p by interpolation.

This does not immediately apply to the irregular case, which is characterized in that a volume element 50 of the subdivision model considered, and in particular a cell 52 thereof, which contains the location to be evaluated, does not define a regular 4 4 4 network, for example because it has extraordinary nodes 12 and / or includes extraordinary edges 14. An important finding, on which the following invention is based, is that such irregular elements 50 can nevertheless be evaluated with a justifiable computational effort, taking into account an alignment of the irregular edge. In particular, it was recognized that such elements 50, at least if they only have an irregular edge, are only irregular in two dimensions after a further subdivision step and are regular in the third (for example irregular in u and v and regular w). Such a structure of a volume element 50, which is only irregular in two dimensions, is also referred to below as a layered structure. If a plurality of irregular edges are present, the necessary number n of subdivision steps must be determined using FIG. 9 explained below until the layered structure is obtained, these subdivisions by calculating the intrinsic structure of the subdivision matrix in a constant calculation time

(arithmetically) can be carried out.

In the following, it is assumed that this is the case that there is no layered structure and shows how to do it in a constant

Calculation time can reach this. However, since structures which have already been layered can also be assumed according to the invention, these measures are not mandatory.

To carry out subdivision steps, numbering or, in other words, indexing of the control points is preferably carried out, so that the topology of the control point network containing the evaluating cell for each

Subdivision step remains consistent. In this connection, the inventors initially recognized that, in contrast to the two-dimensional case, no numbering only depending on a valence of the control point, especially since the topology in the vicinity of the irregular point can be arbitrary. Instead, starting from the irregular edge 15 and with reference to an irregular control point 18, which is to be selected according to the rules explained below with reference to FIG. 7b, the numbers are assigned in the following order, as is also clear from FIGS. 13a-g (ie the checkpoint-specific ones)

Numbers and indexes are assigned to the individual control point groups in ascending order according to the following order). In these Figures 13a-g, the irregular edges are shown in dashed lines.

1. The local coordinate system of the cell is shown in FIG. 13a and the

 selected irregular control point, numbered 0. Irregular, dashed edges with valence three run from this in all three coordinate directions u, v and w. As shown in Figure 13b, all control points are numbered in ascending order, starting from this point by traversing a single edge in the one containing the irregular cell

 Checkpoint network (also called local checkpoint network) can be reached or, in other words, objected to by an edge or an edge length of the irregular edge. These control points can also be called direct neighbors of the extraordinary edge and are numbered 1 to 3 in FIG. 13b. Metaphorically speaking, points 1 to 3 share a common edge with irregular control point 0.

2. Beginning with the number following the last checkpoint of group 1, all checkpoints are numbered in ascending order, which are made by traversing two consecutive or adjacent edges in the checkpoint network containing the irregular cell (also local

Checkpoint network) can be reached or, in other words, objected to by two edges or edge lengths of the irregular edge (see FIGS. 4 to 10 in FIG. 13c). Furthermore, these control points assume that they share an area with the irregular edge (that is, they delimit or can be assigned to a common area with the irregular edge). 3. Beginning with the number following the last checkpoint in group 2, all checkpoints are numbered in ascending order, which are made by traversing three consecutive or adjacent edges in the checkpoint network containing the irregular cell (also local

 Called control point network) can be reached or, in other words, are objected to by three edges or edge lengths of the irregular edge (see FIGS. 11 to 14 in FIG. 13d). Furthermore, these control points assume that they share a cell with the irregular edge (that is, they delimit or can be assigned to a common cell with the irregular edge).

4. In FIG. 13e, the control points of the lower layer are then numbered 15 to 21. The control point of this layer lying along the w axis from FIG. 13 a, viewed on a line with the control point 0, is numbered 15. Based on this, the points surrounding this point 15, which are not necessarily connected directly via an edge, are numbered 16 to 21. Metaphorically speaking, starting from point 15, the surrounding points are numbered in a star shape.

5. In FIG. 13f the control point of the right (outer) layer lying along the v-axis on a line with the control point 0 is numbered 22.

 Proceeding from this, those surrounding this point 22 are not

 necessarily numbered 23 through 31 directly connected to an edge. Metaphorically speaking, starting from point 22, the surrounding points are numbered in a star shape.

6. In FIG. 13g, a control point of the rear (outer) layer lying along the u-axis on a line with the control point 0 is numbered 32.

 Proceeding from this, those surrounding this point 32 are not

 necessarily numbered 33 through 45 directly connected to an edge. Figuratively speaking, starting from point 32, the surrounding points are numbered in a star shape.

To maintain consistency when performing subdivision steps, it is necessary to maintain the index of each control point and its corresponding subdivided control point. Under the correspondence in this It should be understood that the arrangement of the control points around the irregular edge remains constant or similar in a local subdivision. Although new control points are created in the subdivision, these have the same arrangement / structure as that with respect to the (also subdivided) irregular edge

Exit checkpoints before subdivision. Checkpoints before and after the subdivision that have or define analog structures can therefore be described as corresponding to each other.

Assuming that the control point network only comprises hexahedral cells, the following can also be assumed for the control points numbered in the order 1 to 3:

For each control point in the local control point network that shares an edge, area or cell with the irregular edge, the corresponding control point in the subdivided control point network is accordingly an edge point

 Surface point or a cell point, depending on which element is shared with the irregular edge. The terms edge point, surface point and

 Zellpunkt derive from the conventional subdivision rules for Catmull-Clark surfaces or volumes, which were discussed above and for which corresponding quotations are also mentioned. An edge point is a new point that is created when you perform a subdivision step on an edge to subdivide it. The same applies to area and cell points. For each cell e.g. creates a cell point in the arithmetic center of the cell when subdividing according to the Catmull-Clark rules.

The weights of the corresponding subdivided control points are in row i of the subdivision matrix, where i is the assigned index of the

 original checkpoint in the local checkpoint network. The

 Using weights is also conventional

 Subdivision rules for Catmull-Clark surfaces or volumes, which were discussed above and for which corresponding quotations are also mentioned.

The block of the subdivision matrix, which the

Checkpoints in the immediate vicinity of the irregular checkpoint. Precise formulations for subdivision rules and for general layouts of subdivision matrices can be found in the aforementioned scientific articles by Joy and MacCracken. It also contains the Catmull-Clark rules, as mentioned above.

Then the remaining control points are considered that do not share an edge, surface or cell with the irregular edge. These control points form the three outer layers of the control point network, each layer forming a two-dimensional control point network. This can be seen in FIG. 8b, in which a lowermost layer 56a (the base layer in FIG. 8b), a frontal layer 56b, which in the figure directly faces the viewer, and an outer layer 56c on the right in the figure, are shown as outer layers are. The lowermost and frontal layers 56a, b each comprise a control point 18 with the valence 5 (see in each case marked irregular points 18 with a corresponding valence for these layers 56a, b). For these layers 56a, b, one can proceed as in the two-dimensional case from FIG. 4 and one

Numbering according to a distance from the irregular point 18. The right layer 56c does not include an irregular checkpoint and defines a regular two-dimensional checkpoint network.

The corresponding lines in the subdivision matrix can then be generated for the control points 12 of the outer layers 56a, 56b in a conventional manner. It must be taken into account that these three outer layers each share four checkpoints 12 in the exit checkpoint network and five in the subdivided checkpoint network in order to prevent these checkpoints from being taken into account twice. Furthermore, the considered (local) control network along the neighborhood relationships (which is also referred to as local

Neighborhoods or topological information) is traversed, e.g. using a volumetric data structure created for this purpose. When creating the numbering described above, this information is taken into account and the control network is traversed based on the neighborhoods.

For each row of the subdivision matrix, the weights and their corresponding column index can be determined by examining the local neighborhood of the corresponding control point 12. The control points 12, which influence the position of a divided control point 12, can be used by using the topological Information is determined that is contained in the volumetric data structure. The topological information can, for example, indicate which control points are connected to one another or which share a surface, cell or edge.

Finally, by assigning the corresponding control points 12 to their corresponding index in the current control point network 12, the column index is given the corresponding weights.

Since, with the correspondingly numbered subdivision matrix, no changes of direction explained below are required when performing computational subdivisions, the diagonal matrix can be used to carry out the subdivisions, which contains the eigenvalues, as described above in

Connection with the equations (5) - (7) explained.

To compute the eigenstructure, the M x K subdivision matrix is converted into a square matrix by removing the last M - K (ie M less K) rows, where K is the absolute number of control points in the original and M is the absolute number of control points is in the divided checkpoint network. The removed rows correspond to the three outer layers of the divided checkpoint network. Taking into account that these layers are five each

Sharing control points with each other, the number of control points in these layers is:

MK = 2 (Ni + N ₂ + N ₃ ) + 37, where Ni is the valence of the irregular edge that is adjacent to the i-th outer layer. For the control point network in FIG. 8b, for example, N _I = N ₂ = 5 and N ₃ = 4.

In the following, concrete examples of checkpoint networks to be evaluated are discussed.

FIG. 7b shows an example of a volumetric control point network 10 which has irregular edges 15 with a valence of three. This applies in particular to the central cell 52, which contains the location to be evaluated. You can see immediately that there is no regular neighborhood for this cell 32, as shown in FIG. 6.

The example shown can be understood as a volumetric version of the irregular two-dimensional surface network shown in FIG. 7a. In contrast, FIG. 8b shows an example of a volumetric network that has irregular edges 15 (and in particular a central cell 52 with such an edge 15) with a valence of five. This example can be understood as a volumetric version of the irregular two-dimensional surface network shown in FIG. 8a.

As mentioned, in the examples from FIGS. 7b and 8b, an evaluation is to take place at a location which is in the inner central cell 52 and for which there is a new one

Volume element 50 is defined in the form of a local neighborhood around the inner cell 52. In the state shown, the volume element 50 or this neighborhood does not yet represent a regular arrangement of 4 x 4 x 4 control points 12 or 3 c 3 c 3 cells. This is particularly the case because the cell 52 still contains an irregular point 18 .

In order to further subdivide (computationally) the network of the irregular element 50 and in particular the irregular cell 52 from FIG. 7b, so that a layered structure which can be evaluated according to the following equation is obtained, one of the irregular control points 18 contained therein must first be selected. This is one of the end points of an irregular edge 15 which is encompassed by the cell 52, and preferably that irregular control point 18 at which several or all of the irregular edges 15 converge. More specifically, in the case of a plurality of irregular edges 15 which delimit a common cell 52, the irregular control point 18 at which all irregular edges 15 of the cell 52 converge is generally selected. If, in the case of a plurality of irregular edges 15, no such control point 18 can be found at which all irregular edges 15 meet, further subdivision steps can be carried out until this state is reached. In general, it should be noted that an edge 14, 15 is defined in the example shown in such a way that it extends between two control points 12 but not through a plurality of control points 12. The layered structure can then be obtained based on the above rules for defining a subdivision matrix and this can then be selected further. A u, v, w coordinate system 57 for describing locations in the cell 52 is again defined in FIG. 7b. It can be seen that the irregular edge 15 extends along the w axis of this coordinate system 57. This is achieved by first determining the orientation of the irregular edge 15 in the corresponding coordinate system 57 in a first step. This can be determined from the data record defining the subdivision model (or also by manual user input). Furthermore, a preferred axis (in this case w) is specified, along which the irregular edge 15 is to extend. If a relative deviation from the irregular edge 15 and this axis w is determined, for example in the form of an angle between them, the subdivision model and / or the coordinate system 57 can be rotated relative to one another. A definition of the

Coordinate system 57 from the outset in such a way that such

Relative deviation does not occur. This can also be done arithmetically using corresponding coordinate transformations. Alternatively, without any change in the subdivision model (for example without changing its orientation), the coordinate system 57 can be defined from the outset in such a way that its w-axis runs along the irregular edge 15. The state shown in FIG. 7b represents the end result of such measures. The same also applies to FIG. 8b, the w axis of the coordinate system 57 being inclined analogously to the slightly oblique course of the selected irregular edge 15 which delimits the cell 52 to be evaluated (in of the merely schematic FIG. 8b may not be clearly recognizable). It should also be noted that the planes extending orthogonally to the irregular edge 15 in u and v are referred to below as individual layers 56 of the volume element 50.

Analogous to the two-dimensional case and as explained there for example with reference to FIG. 3, the cell 52 to be evaluated must also be locally subdivided in the three-dimensional case, since it initially contains the irregular point 18 and thus cannot be evaluated in the usual way by means of basic functions (ie not according to

Equation (15) can be evaluated). FIG. 9 explained below is used for the corresponding subdivision. In general, the Catmull-Clark subdivision rules, which are used to further subdivide the irregular cell 52, can be summarized in a subdivision matrix (or subdivision matrix) with the size 5M x 4K, at least if there are already layered structures, where K is the number of control points 12 per layer 56 of the control point network 10 or volume element 50 and M the

Checkpoint number of the correspondingly divided network corresponds to 1 1. It should be noted that these numerical values are only correct if a layered structure is already assumed and otherwise the above alternative formulation of the M x K matrix would have to be used. For each additional local subdivision (or, in other words, for each additional local subdivision), eight new sub-elements are created in the form of small-volume cells 52, starting from the cell 52 shown in FIG. 7b.

In the two-dimensional case, an additional local subdivision leads to the fact that all additionally obtained patches become regular, with the exception of the one that contains the irregular point (see respective regular patches k = 1, 2, 3 in FIG. 3 and the remaining patches that correspond to the irregular point included at 0.0). In the volumetric case, the additional local subdivision results in six sub-elements (or cells 52) that become regular, while two irregular cells remain that still contain the irregular edge 15 or adjoin it. Additional subdivisions of these irregular sub-elements would increase the volume fraction of the regular cells, but not to a complete one

Dissolution of the irregular cells 52 lead.

The relationship between the number of additionally required subdivision steps n = 1 ... 4 and the resulting sub-elements k = 1 ... 8 depending on the parameters u, v, w of a location to be evaluated is shown in FIG. 9, n also being significantly larger Accept values and can in principle also be infinite. Figure 9 shows the volumetric variant or volumetric extension of Figure 3.

Alternatively, the formula cited in the general description section can be used. In FIG. 9, the irregular control point 18 is at the position 0, 0, 0. The irregular edge 15 extends from this along the w-axis. This assignment is achieved by the definition of the coordinate system 57, which is based on the previously determined alignment of the irregular edge 15. An at least theoretical possibility to evaluate any location in an irregular cell 52 would be this or its neighborhood in the form of the

Subdivide volume element 50 until the location to be evaluated lies within a completely regular cell 52. If a location to be evaluated is located near an irregular edge 15, but not close to an extraordinary point 18, the direction of the subdivision would have to be changed during the evaluation process.

The direction of the subdivision, which can also be referred to as the subdivision direction, is generally defined by the values of the target parameters u, v, w and is usually (0, 0, 0). This direction does not indicate a direction in the strictly geometrical sense. Instead, this applies to the case that a subdivision “in this direction” takes place for a location in FIG. 9 as a result of iterative subdivisions towards an irregular edge which runs along the w coordinate. In the event that a location to be evaluated is close to an irregular point 18 but away from the irregular edge 15 (for example a location with the coordinates (u, v, w) = (0.1; 0.1; 0.33 )) the direction of the subdivision has to be changed several times in order to achieve a local regular structure. With previous solutions, such as from Burkhart et al. A corresponding change of direction must always take place as soon as the location is in the range k = 3 from FIG. 9. When changing direction, however, it is no longer possible to potentiate the diagonal eigenvalue matrix, but a new matrix expression must be created, which increases the calculation time and generally leads to inconsistent calculation times. The approach presented here prevents this and prevents changes of direction in general, since layered structures can be evaluated directly. Even without a change of direction, the solution presented here enables a layered structure that can be evaluated to be obtained. Through the above

Numbering scheme of the subdivision matrix can also maintain the intrinsic structure of the subdivision matrix and a constant calculation time can be achieved by exponentiating the eigenvalue matrix.

In summary, changes in direction increase, as is also the case with the aforementioned scientific work by Burkhart et al. are used, the required calculation effort and in particular lead to the fact that the

Calculation time with the required additional subdivision steps in Increases substantially linearly (ie is not constant). In addition, information can be lost that makes it difficult or even impossible to transform the subdivided control point network 10 back into its initial state. For this reason, an alternative evaluation approach for irregular volume elements 50 including cells 52 is explained here, which is based, among other things, on a subdivision matrix with the numbering scheme explained above.

10a-b show a representation of the irregular volume elements 50 from FIG. 7b and FIG. 8b after a further local division of the model (ie a division of its volume element 50) (in FIG. 10a the network from FIG. 7b and in FIG. 10b the Network from Figure 8b). The case k = 3 from FIG. 9 is assumed here. The control point networks 10 of the respective volume elements 52 each consist of four layers 56, which each extend in the u-v plane (i.e. orthogonal to the irregular edge 15), and are strung together or, in other words, layered. Furthermore, the inner cells 52 in which the location to be evaluated is located are also marked. Instead of dividing up to a state in which the location to be evaluated lies in a completely regular 4 x 4 x 4 checkpoint neighborhood, in the solution shown it is sufficient to subdivide until the irregular point 18, but not the irregular one Edge 15 lies outside of the cell 52 which comprises the location to be evaluated.

It can be seen that within a respective uv layer 56 an irregular control point 18 with the valence of three (FIG. 10b) or five (FIG. 10a) is mainly contained at the same position, so that the corresponding layers 56 are each irregular . In the w dimension, however, the respective elements are regular, since 4 c 4 control point networks can be defined there.

This takes advantage of the solution disclosed herein in that the bivariate subdivision basis functions (p (u, v) are calculated according to equation (1 1) of the two-dimensional case above and multiplied by a regular B-spline basis function Ni (w). The resulting trivariate basis function is:

Here, j denotes the respective control points in the i-th u, v layer 56. In each layer 56, the valence of the extraordinary edge 15 is equal to the valence of the extraordinary point 18. This enables the use of those previously used only for the two-dimensional case Basic functions even in the three-dimensional case. These basic functions can be used, for example, in equation (15) to determine or evaluate any location e with the coordinates u, v, w.

Likewise, based on these basic functions and the intrinsic structure of the subdivision matrix (set up according to the above numbering scheme), which is contained in the form of an expression analogous to equation (7), an evaluation of a location p can be carried out based on the following expressions:

Pk is a picking matrix, which is analogous to the

two-dimensional case for k = 0, ..., 6 is defined (see also equation (8)). Of the

Expression <p (u, v, w) is also referred to herein as a trivariate subdivision basis function (or trivariate division basis function). The basic function N from equation (17) can also be found in equation (18), but there as a function of

transformed u, v, w coordinate system as well as the necessary subdivision steps n and the area k in which the location to be evaluated lies after n subdivision steps. The above equations (18) - (19) represent equations for the volumetric case which are analogous to the equations (9) - (1 1) of the two-dimensional case, but enable

Evaluations in constant calculation time. Equation (18) contains the complete process described here for evaluating or calculating the limit point p, including setting up the subdivision matrix and its own structure, determining the number of subdivision steps n required and the ranges k etc.

It is understood that the described assignment of the w axis to the irregular

Edge 15 is merely an example and that any of the other u-v axes or axes with entirely different names could also be assigned to the irregular edge 15. The assignment of the u, v, w parameters in equation (17) would then have to be adapted accordingly.

In summary, an evaluation is thus made possible in which a layered structure is first generated by subdivisions without changing direction and thus a constant calculation time, with additional local subdivisions based on the eigenvalue diagonal matrix of the two-dimensional case, which can be made in the eigen basis function term, for this structure (p (u, v) is included. Thus, again at least parts of the relevant matrix expressions are calculated in advance and the computing time is largely determined by the essentially time-constant exponentiation of the corresponding diagonal matrix. As described, this is made possible by first determining the orientation of the irregular edge 15 and selecting a local u, v, w coordinate system in order to obtain layered structures and then by the eigen-base function term (p (u, v) Above, only the evaluation according to the invention of an inner cell 52 has been considered, but cells 52 and volume elements 50, which include interfaces 32 to the environment, represent a special case.

Boundary elements 33 (a boundary element 33 comprises a plurality of cells 52, at least one of which is a boundary cell 34) are to be distinguished from the cases shown in FIGS. 1 a-e. More precisely, two first examples of border cells 34 are shown in FIG. 1 a and 11 b, and a corresponding one in FIG

two-dimensional configuration. 1 1 d and 1 1 e show further examples of borderline cases, for which a corresponding two-dimensional configuration is shown in FIG. 1 1 f.

FIG. 1 a shows a first borderline case, in which an inherently regular structure of a single border layer 32 in the form of the topmost layer 56 in this figure has (the remaining edge surfaces of the border element 33 shown adjoin surfaces of adjacent volume elements 52, not shown, and therefore do not form a corresponding boundary layer 32). In the trivial (ie three-dimensional) case shown, the boundary layer 32 can generally be treated similarly to sharp edges in the bivariate (ie two-dimensional) case, which is why the boundary layer 32 in the corresponding two-dimensional configuration from FIG. 11 c is shown as a corresponding sharp edge 17 .

In particular, FIGS. 1 1 c and 1 1 f show that, analogously to the case of FIGS. 5a, 5b, the sharp edges 17 can be extrapolated in order to define the regular 4 × 4 (for the analog two-dimensional case shown) To add neighborhood missing control points 12 (see correspondingly shown control points 12 and edges in Figures 1 1 c, 1 1 f). Such an actual addition of control points 12 by extrapolation does not have to take place, but legitimizes the use of the sharp-edge basic functions. In order to evaluate a location in the boundary cell 34 shown in FIG. 1 a, which forms part of the boundary layer 32, the B-spline basic functions from the above

Equation (16) replaced with sharp-edge basis functions in the w direction, since there the edge or border area to the environment is present. Each of the three individual sharp edge functions from equation (14) corresponds to one of the layers 56 of the

Checkpoint network from Figure 1 1 a, wherein the first function from equation (14), which begins with the value 1, corresponds to the boundary layer 32. The new border base functions are as follows:

The second borderline case, which is shown in FIG. 11b, is characterized in that an irregular edge 15 runs orthogonally to the border layer 32. Again, a coordinate system 57 is defined such that its w-axis runs along the irregular edge 15 and the u-v plane parallel to layers 56 of the shown

Boundary element 33 runs orthogonal to this edge 15. For one

Evaluation of a location in the boundary cell 34, one edge of which coincides with the irregular edge 15, irregular two-dimensional subdivision basis functions in the u and v dimension are considered and combined with the sharp edge basis functions in the w dimension. The resulting limit basis functions Bi (u, v, w) are as follows, where i and j are defined analogously to equation (17) above:

Bi j (u, V, w) = ipfi III V) Ci (w) i 1, 3, j-1, K (21).

In the third borderline case from FIG. 11d, an irregular edge 15 extends within or along an interface. In other words, the irregular edge 15 is contained in a corresponding boundary layer 32. In FIG. 11 d, this relates to the vertical irregular edge 15 provided with a corresponding reference symbol, which runs in an interface 36 facing away from the viewer. The corresponding two-dimensional case is shown in Figure 1 1 f. The coordinate system 57 of the boundary cell 34 containing the location to be evaluated, which applies to both of FIGS. 1 1 d and 1 1 e, is again defined in such a way that the w axis runs along the irregular edge 15. Furthermore, the layers 56 of the boundary elements 33, which run orthogonally to this edge 15, extend parallel to the uv plane of this coordinate system 57.

In the case of Figure 1 1 d, a solution is obtained by using the

Subdivision basis functions in u and v are calculated and multiplied by the regular B-spline basis functions in the w dimension, which corresponds to equation (21) above. More precisely, the two-dimensional basic functions including sharp edges for the irregular surface piece, as shown in FIG. 1 1 f, are calculated and multiplied by the function in the w dimension.

In the fourth borderline case, which is shown in FIG. 11e, the border cell 34 to be evaluated is positioned in such a way that two of its surfaces within an interface 36 or

Boundary layer 32 of the boundary element 33 lie. These are the upper boundary surface 36 marked in FIG. 1 e and the boundary surface 36 facing away from the viewer, each of which contains corresponding surfaces of the boundary element 33. In this case, the irregular edge 15 is defined as an edge divided by the two interfaces 36 of the boundary cell 34 facing away from the viewer and the irregular point 18 as an end point of this irregular edge 15.

Subsequently, with the aid of the subdivision rules from FIG. 9 and with knowledge of the exact u, v, w coordinates of the location to be evaluated, the number of subdivision steps n is defined, which result in a cell 34 containing the location to be evaluated and obtained after a corresponding subdivision still contains a single surface in a corresponding interface 36 of the boundary element 33.

The size of the Catmull-Clark subdivision matrix is reduced to 4M x 3K, where M corresponds to the number of control points 12 in each layer 56 of the subdivided network 11 and K corresponds to the corresponding number in the output network 10. The reduction is due to the fact that a number of

Control points 12 with respect to a cell arranged in the interior of a volume are missing. Subsequently, one of the first to third limit cases explained above can be used and the relevant location can be evaluated directly accordingly. If a cell 34 to be evaluated has more than two surfaces which lie in boundary surface 36 or two such surfaces but which do not adjoin one another, the output network 10 of the associated boundary element 33 must be subdivided once. A resulting cell 34, which contains the location to be evaluated, then automatically falls under one of the above first to fourth limit cases and can be evaluated accordingly.

In summary, the solution shown thus also provides options for evaluating the subdivision model in its border or border areas in a constant time, so that in fact any locations in the entire model with a constant

Calculation time can be evaluated.

Figure 12 is an illustration to illustrate advantages of the solution disclosed herein. A first straight line 106 and a second straight line 108 are shown in detail, each of which shows a relationship between a required number of subdivision steps (i.e.

additional local subdivisions in order to be able to carry out an evaluation at any location) and represent a time required for this. The time is defined as the relative time based on the time standardized with 1, which is required for an evaluation with the solution according to the invention. In this case, an output network 10 is assumed which, according to the fourth limit case above, comprises a cell 34 with at least two areas in the interfaces of the volume element 52 under consideration and an irregular point 18 which has a valence of five.

The first straight line 106 results from the application of a solution algorithm according to the prior art and more precisely from Burkhart et al. (Daniel Burkhart, Bernd Hamann and Georg Umlauf: Iso-geometric Finite Element Analysis Based on Catmull-Clark: Subdivision Solids In Computer Graphics Forum, Vol. 29.Wiley Online Library, 1575-1584). In the case of locations to be evaluated in irregular cells and / or volume elements, further local subdivisions are often carried out there until the location to be evaluated ultimately lies in a completely regular range.

Accordingly, the computational effort increases linearly with the number of subdivision steps required. Line 108, however, results from the application of the solution disclosed herein. It can be seen that the calculation time is independent of the number of subdivision steps n and is therefore constant, and is also shorter from approx. Three subdivision steps than in the approach according to Burkhart et al. The closer to an irregular point or edge, the greater the advantage.

The constant and shorter at the latest from three subdivision steps

The calculation time primarily results from the fact that for a given output network, essential matrix expressions such as the subdivision matrix and a calculation of its own structure are calculated in advance (i.e. before carrying out evaluations) and e.g. be stored in an electronic storage unit, and furthermore from the fact that one does not have to multiply n and with each subdivision step increasing subdivision matrices, but only potentiates the diagonal matrix. The calculation can be carried out using a computer, in particular one

Computer device, which can also include the aforementioned electronic storage unit. Subsequently, as explained above, only a diagonal matrix has to be potentiated depending on the number n of subdivision steps to be carried out, but this is characterized by generally short and, above all, essentially unchangeable calculation times.

As shown, a

Coordinate determination and / or property determination for any location of a subdivision model take place, in particular also if the basic functions define or specify material properties at individual control points.

Additionally or alternatively, the described solution can be used in particular in the context of simulations, e.g. when properties (especially physical properties) of an object that has been modeled using a volumetric subdivision model are to be simulated. Based on this, the construction of the object can be selected and / or adapted appropriately.

One example is a simulation based on the finite element method (FEM). Forces which act in the model or at the evaluated locations can be determined hereby in a manner known per se. The method can therefore also generally Step of performing a simulation based on the subdivision model and / or determining forces at the location to be evaluated, which can be done in particular on the basis of an FEM.

In a manner known per se, finite element methods investigate displacements of finite elements into which a given object has been divided. The resulting strains and stresses (and consequently also locally acting forces) are then determined from these displacements using a stiffness matrix likewise determined on the basis of the division. For a volumetric subdivision model, each cell can be understood and simulated as a finite element.

A displacement function u for individual control points i can therefore be based on the basic functions N defined in the respective control points, in accordance with the following

Context are defined:

In this case, too, the basic functions N can be evaluated at any location and with a substantially constant calculation time on the basis of the above approaches, and the simulation and force determination can thus be carried out in a particularly computationally efficient manner.

Claims

Claims

1. Method for computer-assisted evaluation of a volumetric

Subdivision model, which is described by a control point network (10) consisting of control points (12) connected by edges (14),

the method comprising:

 Obtaining information regarding one in the volume of the

 Subdivision model lying to be evaluated;

 Determine whether the location to be evaluated lies in a volume element (52) which has an irregular edge (15), which is an edge (15), the one

 Control point (12) connects to one of six different valences, and if so:

 Determining an alignment of the irregular edge (15);

 Evaluation of the volumetric subdivision model at the location to be evaluated based on the determined orientation.

2. The method according to claim 1,

further comprising:

 Defining a coordinate system (57) in accordance with the orientation of the irregular edge (15), in particular such that the irregular edge (15) extends along an axis (u, v, w) of the coordinate system (57).

3. The method according to claim 1 or 2,

further comprising:

 Identify a cell (52) of the subdivision model that the

 contains the location to be evaluated; and

 Determine the number of subdivisions of this cell (52) that are required so that the location to be evaluated is located in an evaluable cell (52) that results from the subdivision of the original cell (52).

4. The method according to claim 3,

the evaluable cell (52) being distinguished by the fact that it has none

Control point with one of six different valences included.

5. The method according to any one of the preceding claims,

the location to be evaluated is evaluated using subdivision basis functions (f).

6. The method according to any one of the preceding claims,

wherein if the volume element does not have a regular structure in at least one dimension, the volume element is subdivided by means of a subdivision matrix and the entries in the subdivision matrix are dependent on at least one structural property of the control point network (10).

7. The method according to claim 6,

wherein the structural property is a position of a checkpoint (12) with one of six different valences within the checkpoint network (10).

8. The method according to claim 7,

wherein if the cell (53) with the location to be evaluated is bounded by a single irregular edge (15), a control point (12) is selected as the control point (12) for specifying the structural property and the entries of the division matrix The end point of the irregular edge (15) is.

9. The method according to claim 7,

wherein if the cell (52) with the location to be evaluated is delimited by a plurality of irregular edges (15), a control point (12) is selected as the control point (12) for specifying the structural property and the entries of the division matrix is connected to all irregular edges (15).

10. The method according to claim 9,

if there is no checkpoint (12) connected to all irregular edges (15), the checkpoint network (10) is subdivided once.

1 1. The method according to any one of the preceding claims, wherein the volumetric subdivision model is also evaluated as a function of whether the location to be evaluated is in a border area of the subdivision model to the surroundings.

12. The method according to claim 1 1,

when the location to be evaluated is in a border area of the

Subdivision model to the environment, at least one base function used for the evaluation is determined based on a sharp edge function (c) along the axis along which the irregular edge (15) also runs.

13. The method according to claim 1 1,

Such an evaluation takes place when it is determined that the irregular edge (15) is orthogonal to or in an interface (32) of the subdivision model to the surroundings.

14. Computer program product,

comprising program instructions for executing a method according to one of the preceding claims when executing these program instructions on a processor unit.

15. computer device,

comprising a processor unit and a memory unit in which a

Computer program product according to claim 9 for execution by the

Processor unit is stored.