CN112581624A - Two-dimensional finite element mesh subdivision algorithm based on distance function defined boundary - Google Patents
Two-dimensional finite element mesh subdivision algorithm based on distance function defined boundary Download PDFInfo
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Abstract
The invention relates to the field of electromagnetic field finite element numerical analysis, in particular to a two-dimensional finite element mesh subdivision algorithm based on a distance function defined boundary. The method comprises the following steps: the distance function constructing step is mainly divided into two cases, regular shapes are expressed by geometric functions, irregular shapes abstract boundary contours in an image processing mode, and boundaries are smoothed by a boundary smoothing algorithm based on path optimization. A Delaunay triangulation step, wherein the topology of the truss is determined by the Delaunay triangulation for use in the mesh smoothing step; and adding a quality evaluation function to limit the grid iteration times. And a mesh smoothing step, namely improving the mesh quality and efficiency through a force-based mesh smoothing algorithm. And the post-processing module is used for screening different medium boundaries based on a distance function and a second-order Gaussian projection method, giving different boundary excitation conditions according to different application scenes, and further solving to obtain a result.
Description
Technical Field
The invention relates to the field of electromagnetic field finite element numerical analysis, in particular to a two-dimensional finite element mesh generation algorithm based on a distance function defined boundary.
Background
The method has excellent capability of adapting to the field boundary geometric shape and the medium physical property variation, so that the finite element method becomes one of the dominant numerical calculation methods for the quantitative analysis and the optimized design of various electromagnetic fields and electromagnetic wave engineering problems. In general, finite element analysis is divided into: establishing a model (preprocessing), calculating and solving, processing a result and evaluating (postprocessing). Wherein, the time ratio of each stage is respectively as follows: 40-60%, 5-10% and 30-50%. Therefore, how to disperse a complex field in actual engineering into a 'finite element', the subdivision problem is an extremely critical step in a numerical analysis process, and the accuracy and the effectiveness of a final result are directly determined.
Grid generators, such as the large business software CAE/Maxwell, are based on the mapping method as a theoretical basis. The method has the main idea that a geometric domain to be divided is mapped to a parameter domain of a parameter space forming rule through a proper mapping function; mesh generation is carried out on the regular parameter domain; the meshes of the parameter domain are inversely mapped back to the geometry domain, resulting in finite element meshes of the geometry domain. The mapping method is a non-full-automatic method, and a subdivision object must be firstly subdivided into sub-domains with simple topological relations in a manual interaction mode. These commercial software using mapping as a core algorithm are expensive, complex to operate, slow in generating large-scale triangular meshes, and hardly accessible, and they are often used as "black boxes" and difficult to interact with other code or software, so users give up control. The ability to understand and use the mesh generation algorithm is a very valuable option and deserves further investigation.
Disclosure of Invention
1. Technical problem to be solved
The purpose of the invention is: the invention provides a two-dimensional finite element mesh generation algorithm based on a distance function defined boundary, which is an unstructured mesh generation algorithm based on the distance function defined boundary, wherein a smooth function with attractive repulsive force effect is used for mesh iteration optimization, and the efficiency of a calculation process and the precision of a calculation result are improved.
2. Technical scheme
In order to solve the above problems, the present invention adopts the following technical solutions.
A two-dimensional finite element mesh subdivision algorithm based on a distance function definition boundary comprises the following steps:
step 1: constructing a distance function, wherein a regular shape is represented by a geometric function, and an irregular shape abstracts a boundary outline by an image processing mode; initialization: generating grids of equidistant points in a boundary frame, giving node coordinates x and y, and storing the coordinates in an array with 2 columns;
step 2: calling a distance function and taking the node position as input, the output is a column vector of distances from the node to the geometric boundary, keeping only the interior points of negative distances, allowing a tolerance geps, then evaluating h (x, y) at each node and comparing 1/h (x, y)2The fixed node array of the user is positioned on the first row of the array, and the duplication of the nodes is removed;
and step 3: the two-dimensional finite element mesh division algorithm enters a main loop, the positions of nodes are iteratively improved in the loop, and the loop is started;
and 4, step 4: performing Delaunay triangulation, removing triangles with centroids outside, rearranging vertices of the triangles anticlockwise to form all edges without repetition, and extracting boundary edges;
and 5: force-based smoothing functions, which are typically done for p and each point movement to maintain the correct topology, are determined by Delaunay triangulation before applying the force-balancing smoothing function; in order to save computation time, when the maximum displacement since the last triangulation is larger than ttol, wherein h is relative to h0And when ttol is preset to be 0.1, re-triangulation is needed:
calculate and combine edge forces and move the node. After N iterations, edges with normalized length greater than 1.5 are segmented by adding midpoints, the Delaunay function generates triangulation of the convex hull of the point set, and triangles outside the geometric figures must be deleted, where a triangle will be deleted if d of the centroid of the triangle is greater than 0;
triangle lists are arrays with 3 columns, each row representing a triangle with three integer indices, each triangle contributing three node pairs when creating an edge list, duplicate entries must be deleted since most pairings will occur twice, the border nodes of the mesh are determined to be the end points of the border edges, and if a point ends outside the geometry after updating the array, it is moved back to the nearest point on the border;
step 6: termination conditions were as follows: calculating the relative displacement of the inner points, and if a large displacement is detected in the current iteration, turning to the step 5; meanwhile, the triangle quality is checked by calculating the triangle area, and if the triangle element quality is more than 0.5, the termination is confirmed;
and 7: screening different medium boundaries based on a distance function and a second-order Gaussian projection method, and giving different boundary excitation conditions according to different application scenes;
and 8: TEA was carried out.
Further, the definition of the geometric domain requires a closed distance function to represent the boundary.
Further, before applying the force balance based smoothing function, determining the topology of the truss by using Delaunay triangulation; in performing the force-based smoothing, the mass of each grid cell will be a proof of adaptation as a force smoothing function.
3. Advantageous effects
Compared with the prior art, the invention has the advantages that:
the invention defines finite element subdivision domain by implicit distance function, the complexity of the calculation process is reduced compared with the mapping method, the invention is a simple domain definition mode, for the distance function which is difficult to be expressed by geometric function, the boundary contour can be identified by image processing, and further the distance function is indirectly defined; compared with commercial software, the method has the advantages that efficiency and precision are greatly improved, and the solution of the user region of interest can be obtained through a grid size adaptive technology.
Drawings
FIG. 1 is a flowchart of the overall algorithm;
FIG. 2 is a schematic diagram of the effect of a smoothing function based on force balance;
FIG. 3(a) is a graph of dispersed nodes within a geometric domain;
FIG. 3(b) is a diagram of Delaunay triangulation of dispersed nodes within a geometric domain;
FIG. 3(c) is a graph after mesh smoothing optimization with a force-based smoothing function;
FIG. 4 is a schematic engineering diagram of a TEAM works protocol 30A three-phase motor;
FIG. 5 is a grayscale image after a graying process;
FIG. 6 is a geometric profile based on a boundary extraction algorithm;
FIG. 7 is a geometric contour map after being processed by a boundary smoothing algorithm based on path optimization;
FIG. 8 is a grid split view of the present invention;
FIG. 9 illustrates the flux density and flux lines at 600rad/s for an exemplary embodiment;
Detailed Description
The present invention uses a signed distance function to specify and describe the geometric region to be gridded. The distance function d (x, y) specifies the shortest distance from any point in space to the boundary of the domain, where the sign of the function is positive outside the region, negative inside, and zero on the boundary. This definition is used to identify whether a point is located within or outside of a geometric domain. Furthermore, the gradient of the distance function points in the direction of the boundary, allowing the outer points to effectively move back into the domain. The geometric domain Ω may be implicitly defined as:
Ω＝{x＝(x,y)∈R2|d(x)＜0} (1)
with boundaries of general geometric background, the present invention formulates a distance function, a simple example of which is a two-dimensional unit circle having a distance function d (r) r-1, where r is sqrt (x 2+ y 2) is the distance to the origin. For more complex geometries, the present invention provides another way to define the solution domain using an image mapping strategy. The invention provides an image boundary extraction algorithm based on path optimization to acquire image boundary information. The algorithm process is as follows: firstly, carrying out gray processing on an image; then carrying out gray level image self-adaptive multi-threshold segmentation (Otsu); and secondly, extracting a polygon boundary according to the gray value, and optimizing the polygon outline by a path optimization algorithm. A specific measure is to recalculate the horizontal coordinate x and the vertical coordinate y of the point p (x, y) by using the formula (2).
Where i is the number of points, t is a proportional value, and the value is between 0 and 1, and p is an N × 2 array of node coordinates. The boundary information obtained by image processing is abstracted into a distance function by using the inprogon similar to MATLAB, and the boundary of the solution domain is described by the distance function.
The internal boundaries must be defined when generating the finite element mesh, which may satisfy the need to solve differential equations with discrete material coefficients. The inner boundary divides the entire finite element solution domain into several sub-domains, which are connected only by common nodes on common edges. The invention proposes to represent the internal boundary phi by a smoothed implicit distance function phi (x). The projection mode is defined by the following formula: higher order approximation projections can be obtained by assuming a smooth implicit function phi approximation estimate, containing more terms in the truncated taylor expansion of phi. For one point (x, y) and one small displacement (Δ x, Δ y), settings are made
In the algorithm proposed by the present invention, the required meshThe size is provided by the mesh density function h (x). The mesh size function h (x) is important for generating high quality mesh objects. The mesh size adaptation function proposed by the present invention is based on the distance from the medial axis, which allows the use of any computational domain defined by a signed distance function. The medial axis is a set of interior points that are equidistant from two or more points on the boundary. The middle axis may be calculated by finding the singularity of the distance function. Calculating intermediate shaft
。
An appropriate mesh smoothing algorithm may optimize the quality of the generated mesh and improve the efficiency of mesh generation. The present invention proposes a new force-based mesh smoothing algorithm that treats the mesh as a spring-like deformable medium. The whole process can be divided into two stages: a first stage, a quality measure of each element and an average quality of the whole grid; and in the second stage, grid smoothing operation based on grid quality.
The node locations are optimized based on a force-based smoothing process, by analogy between triangular meshes and two-dimensional truss structures. The sides of the triangle correspond to the rods and the vertices correspond to the nodes of the truss. The balance problem corresponds to a non-linear equation:
F(p)＝0 (5)
a steady solution is found, and the formula (5) is satisfied. The system uses the forward euler method for approximation. At discrete time tkWhen k Δ t, approximate solution pk≈p(tk) The updating is as follows: p is a radical ofk+1＝pk+ΔtF(pk)。
In the first stage, the invention intends to adopt a method for calculating the quality of a mesh, and in finite element application, the quality measure of a triangular unit mesh can be regarded as the quality measure of a geometric element, and can be calculated in the following way: the upper bound error depends on the smallest angle in the grid. To quantify the grid quality, a commonly used quality metric is the relationship between the maximum inscribed circle radius and the minimum circumscribed circle radius, i.e.
a, b and c are side lengths. According to the rule of thumb, if q of all elementsminAbove 0.5, the mesh is excellent. When the quality of all meshes is greater than 0.5, the mesh generation algorithm will be terminated, which will effectively avoid excessive mesh iterations.
For the second stage, the invention depends on its current length | | | lij| setting each edge to l | |ij＝[pi，pj]Variable spring constant f (p)i，pj). The desired length is eij. The invention adopts the concept of normalized length:
a normalized triangular cell mesh of side length 1 is then created.
Based on (1-s)2)exp(s3-s4) A new smoothing function is constructed. The new Smoothing function NSF (novel Smoothing function) formula (8) constructed by the algorithm has both repulsion and attraction, and in addition, the Smoothing function can accurately limit the force in the Smoothing process through a quality monitor.
q is the element mass of the finite element, qavgIs the average of the entire grid, sij＝//lij///eijAnd// lij// is the current length, the desired length is eij。
The invention adopts a first-order motion equation (9) and p (0) is p0And is approximated by the forward Euler method. At discrete time tkK Δ t, and pi kSet as the position of node i at the kth time step, approximate solution pk≈p(tk) Update to equation (10):
Niis a neighborhood of node i.
As shown in fig. 2, mesh smoothing is performed. When the distance between two vertices in a triangle element of the mesh is too close, s <1, the NSF will exert a repulsive force on the bar formed by the connection of the two vertices; when the distance between two vertices is large, s > 1, NSF will generate a bar-shaped attraction. Since NSF can generate attraction or repulsion according to the value of s, repulsion or attraction on the edges of adjacent meshes will contribute when applied to the entire mesh, and adjacent mesh iteration time can be reduced, which will reduce the overall mesh iteration time. Furthermore, the application of force is limited by the quality of the mesh, which will avoid the generation of a mesh of extremely poor quality.
As shown in fig. 1, a two-dimensional finite element mesh division algorithm based on a distance function defined boundary includes the following steps:
as shown in fig. 3(a), step 1: constructing a distance function, wherein a regular shape is represented by a geometric function, and an irregular shape abstracts a boundary outline by an image processing mode; initialization: generating grids of equidistant points in a boundary frame, giving node coordinates x and y, and storing the coordinates in an array with 2 columns;
step 2: calling a distance function and taking the node position as input, the output is a column vector of distances from the node to the geometric boundary, keeping only the interior points of negative distances, allowing a tolerance geps, then evaluating h (x, y) at each node and comparing 1/h (x, y)2The fixed node array of the user is positioned on the first row of the array, and the duplication of the nodes is removed;
and step 3: the two-dimensional finite element mesh division algorithm enters a main loop, the positions of nodes are iteratively improved in the loop, and the loop is started;
as shown in fig. 3(b), step 4: performing Delaunay triangulation and removing triangles with centroids outside, rearranging vertices of the triangles anticlockwise to form all edges without repetition and extracting boundary edges;
as shown in fig. 3(c), step 5: force-based smoothing functions, which are typically performed for p and each point movement to maintain the correct topology, are determined by Delaunay triangulation before applying the force-balancing smoothing function; to save computation time, when the maximum displacement since the last triangulation is greater than ttol, where h is relative to h0And when ttol is preset to be 0.1, re-triangulation is needed:
calculate and combine edge forces and move the node. After N iterations, edges with normalized length greater than 1.5 are segmented by adding midpoints, the Delaunay function generates triangulation of the convex hull of the point set, and triangles outside the geometric figures must be deleted, where a triangle will be deleted if d of the centroid of the triangle is greater than 0;
triangle lists are arrays with 3 columns, each row representing a triangle with three integer indices, each triangle contributing three node pairs when creating an edge list, duplicate entries must be deleted since most pairings will occur twice, the border nodes of the mesh are determined to be the end points of the border edges, and if a point ends outside the geometry after updating the array, it is moved back to the nearest point on the border;
step 6: termination conditions were as follows: calculating the relative displacement of the inner points, and if a large displacement is detected in the current iteration, turning to the step 5; meanwhile, the triangle quality is checked by calculating the triangle area, and if the triangle element quality is more than 0.5, the termination is confirmed;
and 7: screening different medium boundaries based on a distance function and a second-order Gaussian projection method, and giving different boundary excitation conditions according to different application scenes;
and 8: TEA was carried out.
As shown in fig. 4, the diagram is an engineering diagram of a TEAM works shop layout 30A three-phase motor, and fig. 5 is obtained by filling fig. 4 with equal-difference RGB values; on the basis of gray processing, performing gray segmentation by adopting image self-adaptive multi-threshold segmentation;
extracting the geometric edge of the image in fig. 5 by an image edge detection technique, which is shown in fig. 6; as can be seen from FIG. 6, the extracted geometric boundaries are not smooth enough and may have a limited impact on the accuracy of the finite element analysis;
the geometric edges extracted from fig. 5 are smoothed and optimized by a boundary smoothing algorithm based on path optimization, resulting in the graph shown in fig. 7.
Comparing fig. 7 with fig. 6, it is apparent that the smoothness of the geometric boundary is improved, which helps to improve the accuracy of the mesh generation. Finally, using the mesh generation algorithm proposed in the present invention, a mesh generation diagram is shown in fig. 8. As can be seen from fig. 8, the inner boundary processing method adopted in the present invention is excellent in performance, and the inner boundary is in a smooth state.
Different medium boundaries are screened based on a distance function and a second-order Gaussian projection method, different boundary excitation conditions are given according to the graph 4, and the magnetic flux density and the magnetic flux lines can be obtained when the rotating speed is 600 rad/s.
The above; but are merely preferred embodiments of the invention; the scope of the invention is not limited thereto; any person skilled in the art will be within the technical scope of the present disclosure; the technical scheme and the improved concept thereof according to the invention are equally replaced or changed; are intended to be covered by the scope of the present invention.
Claims (3)
1. A two-dimensional finite element mesh subdivision algorithm based on a distance function definition boundary is characterized in that: the method comprises the following steps:
step 1: constructing a distance function, initializing, generating a grid of equidistant points in a boundary frame, giving node coordinates x and y, and storing the coordinates in an array of 2 columns;
step 2: calling distance function and taking node position as input, and outputting result from node to geometryColumn vectors of distances of the boundary, keeping only the interior points of negative distances, then evaluating h (x, y) at each node and comparing with 1/h (x, y)2The fixed node array of the user is positioned in the first row of the array, and the nodes are also subjected to duplicate removal;
and step 3: the two-dimensional finite element mesh division algorithm enters a main loop, the positions of nodes are improved in an iterative manner in the loop, and the loop is started;
and 4, step 4: performing Delaunay triangulation, removing triangles with centroids outside, rearranging vertices of the triangles anticlockwise to form all edges without repetition, and extracting boundary edges;
and 5: force-based smoothing function: before applying the force balance smoothing function, the topology of the truss is determined using Delaunay triangulation, which is typically done for arrays and each point movement to maintain the correct topology; in order to save the calculation time, when the maximum displacement from the last triangulation is larger than ttol, re-triangulation is needed;
calculating and combining edge forces and moving nodes, after N iterations, dividing edges with normalized length larger than 1.5 by adding a midpoint, generating triangulation of a convex hull of a point set by a Delaunay function, and deleting triangles outside the geometric figures, wherein if the d of the centroid of each triangle is larger than 0, the triangle is deleted;
triangle lists are arrays with 3 columns, each row represents a triangle with three integer indices, each triangle contributes three node pairs when creating the edge list, since most pairings will occur twice, duplicate entries must be deleted, the border nodes of the mesh are determined to be the end points of the border edges, and if a point ends outside the geometry after updating the array, it is moved back to the nearest point on the border;
step 6: termination conditions were as follows: calculating the relative displacement of the inner points, and if a large displacement is detected in the current iteration, turning to the step 5; meanwhile, the triangle quality is checked by calculating the triangle area, and if the triangle element quality is more than 0.5, the termination is confirmed;
and 7: screening different medium boundaries based on a distance function and a second-order Gaussian projection method, and giving different boundary excitation conditions according to different application scenes;
and 8: FEA was performed.
2. The two-dimensional finite element mesh partitioning algorithm for defining a boundary based on a distance function as claimed in claim 1, wherein: the definition of the geometric domain requires a closed distance function to represent the boundary.
3. The two-dimensional finite element mesh partitioning algorithm for defining a boundary based on a distance function as claimed in claim 1, wherein: before applying a smooth function based on force balance, determining the topology of the truss by using Delaunay triangulation; and in the process of carrying out the force-based smoothing function, the quality of each grid unit is taken as a proof that the force smoothing function is self-adaptive.
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Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
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| CN113761762A (en) * | 2021-08-03 | 2021-12-07 | 西北核技术研究所 | Balance flux construction method for finite element numerical simulation posterior error estimation |
| CN114255241A (en) * | 2021-11-16 | 2022-03-29 | 鹏城实验室 | Region segmentation method, device and equipment for path planning and storage medium |
| CN115471635A (en) * | 2022-11-03 | 2022-12-13 | 南京航空航天大学 | Multi-block structure grid singularity identification method based on Delaunay graph |
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2020
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Cited By (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113761762A (en) * | 2021-08-03 | 2021-12-07 | 西北核技术研究所 | Balance flux construction method for finite element numerical simulation posterior error estimation |
| CN113761762B (en) * | 2021-08-03 | 2023-10-20 | 西北核技术研究所 | Posterior error estimation method for electric field/temperature finite element numerical solution |
| CN114255241A (en) * | 2021-11-16 | 2022-03-29 | 鹏城实验室 | Region segmentation method, device and equipment for path planning and storage medium |
| CN114255241B (en) * | 2021-11-16 | 2023-10-20 | 鹏城实验室 | Region segmentation method, device, equipment and storage medium for path planning |
| CN115471635A (en) * | 2022-11-03 | 2022-12-13 | 南京航空航天大学 | Multi-block structure grid singularity identification method based on Delaunay graph |
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