CN109816793B

CN109816793B - Isogeometric analysis-based bas-relief model establishing method

Info

Publication number: CN109816793B
Application number: CN201811494117.4A
Authority: CN
Inventors: 徐岗; 凌成南; 许金兰
Original assignee: Hangzhou Dianzi University
Current assignee: Hangzhou Dianzi University
Priority date: 2018-12-07
Filing date: 2018-12-07
Publication date: 2022-12-06
Anticipated expiration: 2038-12-07
Also published as: CN109816793A

Abstract

The invention discloses a method for establishing a bas-relief model based on isogeometric analysis, which mainly comprises four stages: the method comprises the steps that a user inputs a design sketch, area decomposition and parameterization are carried out on a sketch area, sampling points are selected on a parameterization result according to relief strokes on the sketch, and a low relief model is obtained by solving a Poisson equation with limiting conditions. The invention realizes the acquisition of the low-relief model from the sketch designed by the user, the input form of the sketch can enable the user to freely design the external outline and the internal relief shape of the relief model, and the input form can enable the design of the user to be smoother, thereby getting rid of the model construction mode of the traditional CAD software object button and enabling the design stage to have more freedom and interest.

Description

Isogeometric analysis-based bas-relief model establishing method

Technical Field

The invention belongs to the field of isogeometric analysis application, and particularly relates to a method for establishing a bas-relief model based on isogeometric analysis.

Background

The relief is a combination of sculpture and painting, processes an object by a compression method, expresses a three-dimensional space by elements such as perspective, and is viewed only on one side or both sides. The relief is typically attached to a flat surface or surface, often visible in buildings and vessels. The embossments are classified into low embossments and high embossments according to the difference of compression spaces. The low relief has low rising position and large compression of the body, namely, the engraved pattern only slightly protrudes out of the bottom surface, so that the engraved pattern is more decorative and occupies less area, unlike the high relief with strong relief feeling and space depth, which is formed by obviously protruding the surface of an engraved object. The bas-relief has not only architectural planarity but also a certain sense of volume and relief. Its properties make it more suitable for being attached to a flat carrier than a high relief.

In previous work, bas-reliefs were mostly constructed by the generation and compression of height fields, the proposed method being based on 3D scenes or images and focusing on producing non-linear compression algorithms. The problem that these algorithms mainly deal with is how to keep the depth of the relief model continuous when compressing the height field. Most of the methods calculate depth values through perspective transformation, and reconstruct a relief model by regarding a 3D scene as a height field.

Disclosure of Invention

The invention aims to make the shape and the pattern of a model more creative and enable a user to design more freely, and provides a bas-relief model establishing method based on isogeometric analysis. In the process of establishing the model, an isogeometric analysis method based on a spline theory is adopted, and the isogeometric analysis method is characterized in that the geometric model and the analysis model use the same spline base, so that the same expression of the geometric model and the analysis model is realized. Splines also have the advantage of representing the model with fewer degrees of freedom than triangulated meshes or pixels. The model obtained by the method is continuous on the whole, and the smoothness inside the model can be adjusted according to the multiplicity of the nodes. The invention is mainly divided into four stages: the method comprises the steps that a user inputs a design sketch, area decomposition and parameterization are carried out on a sketch area, sampling points are selected on a parameterization result according to relief strokes on the sketch, and a low relief model is obtained by solving a Poisson equation with limiting conditions.

The method comprises the following specific steps:

step 1, inputting a design sketch.

And 2, fitting the closed boundary of the design sketch by using a least square asymptotic iterative fitting algorithm to obtain a B spline curve boundary.

Step 3, sampling the boundary of the B-spline curve, and sequentially connecting each sampling point to form a polygon; and (4) dividing the interior of the polygon into a subarea set D by applying a subarea division algorithm.

And 4, carrying out Coons surface interpolation on each sub-region unit in the sub-region set D to obtain internal control points of the sub-region unit, so as to obtain plane splines with n m control points corresponding to each sub-region unit, and integrating the sub-region set D into a plane spline set K, wherein n is not less than 2,m not less than 2.

And 5, solving the Poisson equation with homogeneous boundary conditions on the plane spline set K by using an isogeometric analysis method to obtain a base surface.

And 6, if the designed sketch has no internal lines, outputting a base curved surface, otherwise, continuing the following steps: the inner line of the design sketch is expressed by adopting the inner control points and the boundary control points of all the sub-area units, one control point which is closest to each point on the inner line of the design sketch is taken as a characteristic point, and the set of all the characteristic points is marked as L. Then, solving the Poisson equation added with the characteristic line constraint based on the base surface by using an isogeometric analysis method to obtain a bas-relief model.

Further, in step 3, the boundary of each sub-region unit in the sub-region set D is fitted with four splines.

Further, in step 5, the Z coordinate of each point on the base surface is calculated as follows:

designing a closed boundary of a sketch on an XOY plane, wherein S is a Z coordinate of each point on a base curved surface, and XYZ is a Cartesian space coordinate system; Δ is laplace operator; f is a constant, and f > 0;

further, in step 6, the Z coordinate of each point on the bas-relief model is calculated as follows:

wherein, the first and the second end of the pipe are connected with each other,

z coordinates of each point on the bas-relief model;

h is the height that the relief needs to be raised,

the projection of the normal vector of the position of the point needing to be lifted on the base curved surface on the Z axis.

The invention has the beneficial effects that:

the invention really realizes the acquisition of the low relief model from the draft designed by a user. The input form of the sketch allows the user to freely design the outer contour of the relief model as well as the inner relief shape. The input form can enable the design of a user to be smoother, so that a model construction mode of the object button of the traditional CAD software is eliminated, and the design stage has more freedom and interest.

Drawings

FIG. 1 is a flow chart of the present invention;

FIG. 2 (a) is a schematic design drawing;

FIG. 2 (b) is an exploded view of the area of FIG. 2 (a);

FIG. 2 (c) is a control point grid diagram after the area of FIG. 2 (a) is exploded;

fig. 3 (a) is a diagram of a base surface model of fig. 2 (a) (the bas surface model corresponds to a bas surface model that is a stamp, and therefore the height of the base surface model is intentionally set to zero);

FIG. 3 (b) is a diagram of the low relief model of FIG. 2 (a) obtained by the present invention;

FIG. 4 (a) is a schematic diagram of a butterfly design;

FIG. 4 (b) is a diagram of a base surface model of FIG. 4 (a);

FIG. 4 (c) is a diagram of a bas-relief model of FIG. 4 (a) obtained by the present invention;

FIG. 5 (a) is a schematic diagram of dolphin design;

FIG. 5 (b) is a diagram of a base surface model of FIG. 5 (a);

fig. 5 (c) is a diagram of a low relief model obtained by the present invention in fig. 5 (a).

Detailed Description

The invention will be further explained with reference to the drawings.

As shown in fig. 1, the method for establishing the bas-relief model based on isogeometric analysis comprises the following specific steps:

step 1, inputting a design sketch U to be processed, as shown in fig. 2 (a).

And 2, fitting the closed boundary of the design sketch U by using a least square asymptotic iterative fitting (LSPIA) algorithm to obtain a B spline curve boundary.

Step 3, sampling the boundary of the B-spline curve, and sequentially connecting each sampling point to form a polygon; the inside of the polygon is subdivided into a sub-region set D by applying a region subdivision algorithm, the boundaries of the sub-region units are fitted by four spline curves, and the control polygon of the four spline curves in the embodiment is shown in FIG. 2 (b).

And 4, performing Coons surface interpolation on each sub-region unit in the sub-region set D to obtain internal control points of the sub-region unit, so as to obtain plane splines with the number of n × m of the control points corresponding to each sub-region unit, and integrating the sub-region set D into a plane spline set K, wherein n is more than or equal to 2,m is more than or equal to 2, as shown in fig. 2 (c).

The Coons surface interpolation is specifically as follows:

P _i,j represents the ith row and jth column of sub-region units D in the sub-region set D _i,j The internal control point of (1); p _0,0 、P _n,0 、P _0,m 、P _n,m 、P _0,j 、P _n,j 、P _i,0 、P _i,m All represent sub-region unit D _i,j And if the sub-region unit D _i,j If the boundary of (D) is not on the closed boundary of the design sketch, the corresponding sub-area unit D _i,j The boundary control point is obtained by linear interpolation, if the boundary control point is a sub-region unit D _i,j Is located on the closed boundary of the design sketch, the corresponding sub-area unit D _i,j And the boundary control points are taken from the control points of the boundary of the B-spline curve obtained in the step 2. Obtaining each sub-region unit D in the sub-region set D _i,j Then obtaining a sub-area unit D _i,j A planar spline representation of (a);

and 5, solving the Poisson equation with homogeneous boundary conditions on the plane spline set K by using an isogeometric analysis method to obtain a base surface, as shown in FIG. 3 (a).

Designing a closed boundary of a sketch on an XOY plane, wherein S is a Z coordinate of each point on a base curved surface, and XYZ is a Cartesian space coordinate system; Δ is laplace operator; f is a constant and f > 0;

and 6, if the designed sketch has no internal lines, outputting a base curved surface, otherwise, continuing the following steps: expressing an internal line of the design sketch by using internal control points and boundary control points of all sub-region units, taking a control point which is closest to each point on the internal line of the design sketch as a characteristic point, and recording a set of all the characteristic points as L; then, solving Poisson equation added with characteristic line constraint based on the base surface by using isogeometric analysis method to obtain bas-relief model (as shown in FIG. 3 (b)),

wherein the content of the first and second substances,

z coordinates of each point on the low relief model;

h is the height that the relief needs to be raised,

Two further examples of applications are listed below:

the butterfly design sketch shown in fig. 4 (a) shows a base surface model obtained by the method of the present invention as shown in fig. 4 (b), and a final bas-relief model as shown in fig. 4 (c).

FIG. 5 (a) shows a schematic diagram of a dolphin design, and a base surface model obtained by the method of the present invention is shown in FIG. 5 (b), and a final bas-relief model is shown in FIG. 5 (c).

Claims

1. The method for establishing the bas-relief model based on isogeometric analysis is characterized by comprising the following steps of: the method comprises the following steps:

step 1, inputting a design sketch;

step 2, fitting the closed boundary of the design sketch by using a least square asymptotic iterative fitting algorithm to obtain a B spline curve boundary;

step 3, sampling the boundary of the B-spline curve, and sequentially connecting each sampling point to form a polygon; dividing the interior of the polygon into a subarea set D by applying a subarea division algorithm;

step 4, carrying out Coons surface interpolation on each sub-region unit in the sub-region set D to obtain internal control points of the sub-region unit, thereby obtaining plane splines with the number of n × m of the control points corresponding to each sub-region unit, and integrating the sub-region set D into a plane spline set K, wherein n is more than or equal to 2,m and is more than or equal to 2;

step 5, solving a Poisson equation with homogeneous boundary conditions on the plane spline set K by using an isogeometric analysis method to obtain a base surface;

and 6, if the designed sketch has no internal lines, outputting a base curved surface, otherwise, continuing the following steps: expressing an internal line of the design sketch by using internal control points and boundary control points of all sub-region units, taking a control point which is closest to each point on the internal line of the design sketch as a characteristic point, and recording a set of all the characteristic points as L; then, solving the Poisson equation added with the characteristic line constraint based on the base surface by using an isogeometric analysis method to obtain a bas-relief model.

2. The method for building a bas-relief model based on isogeometric analysis as claimed in claim 1, wherein: in step 3, the boundary of each sub-region unit in the sub-region set D is fitted by four spline curves.

3. The method for building a bas-relief model based on isogeometric analysis as claimed in claim 1, wherein: in step 5, the Z coordinate of each point on the base surface is calculated as follows:

designing a closed boundary of a sketch on an XOY plane, wherein S is a Z coordinate of each point on a base curved surface, and XYZ is a Cartesian space coordinate system; Δ is laplace operator; f is a constant and f > 0.

4. The method for building a bas-relief model based on isogeometric analysis as claimed in claim 1, wherein: in step 6, the Z coordinate of each point on the bas-relief model is calculated as follows:

wherein the content of the first and second substances,

z coordinates of each point on the bas-relief model;

h is the height that the relief needs to be raised,