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This paper describes an algorithm to extract adaptive and quality 3D meshes directly from volumetric imaging data. A top-down octree subdivision coupled with the dual contouring method is used to rapidly extract adaptive 3D finite element meshes with correct topology from volumetric imaging data. The edge contraction and smoothing methods are used to improve the mesh quality. The main contribution is extending the dual contouring method to crack-free interval volume 3D meshing with feature sensitive adaptation.
Compared to other tetrahedral extraction methods from imaging data, our method generates adaptive and quality 3D meshes without introducing any hanging nodes. The algorithm has been successfully applied to constructing the geometric model of a biomolecule in finite element calculations. The development of finite element simulations in medicine, molecular biology and engineering has increased the need for high quality finite element meshes. We assume a continuous function F is constructed through the trilinear interpolation of sampled values for each cubic cell in the volume.
For accurate and efficient finite element calculations, it is important to have adaptive and high quality geometric models with minimal number of elements. The studied object may have complicated topology. Figure 21 shows an interval volume between two isosurfaces from the SDF volumetric data of a knee.
The two surfaces have the same topology in Figure 21 d , while the topology of the inner surface may be different from the topology of the outer one Figure 21 b. In this paper, we present a comprehensive approach to extract tetrahedral and hexahedral meshes directly from imaging data. The bilateral prefiltering coupled with anisotropic diffusion methods [ 5 ] is applied to smooth the volumetric data. Accurate gradient estimation can also be obtained. The Contour Spectrum [ 4 ] provides quantitative metrics of a volume to help us select two suitable isovalues for the interval volume.
If the imaging data has no noise for example, SDF data and two isovalues are given to define the interval volume, the preprocessing step can be skipped. We extend the idea of dual contouring to interval volume tetrahedralization and hexahedralization from volumetric Hermite data position and normal information. Dual Contouring [ 31 ] analyzes those edges that have endpoints lying on different sides of the isosurface, called sign change edge.
Each sign change edge is shared by four uniform case or three adaptive case cells, and one minimizer is calculated for each of them by minimizing a predefined Quadratic Error Function QEF [ 25 ]. The QEF is defined as follows:. For each sign change edge, a quad or a triangle is constructed by connecting the minimizers.
These quads and triangles provide an approximation of the isosurface. Each sign change edge belongs to a boundary cell. We present a systematic way to tetrahedralize the volume in the boundary cell. For uniform grids, it is easy to deal with the interior cells. We only need to decompose each cell into five tetrahedra.
The adaptive case is more complicated. In order to avoid introducing hanging nodes, which are strictly prohibited in finite element meshes, we design an algorithm to tetrahedralize the interior cell depending on the resolution levels of all its neighbors.
Figure 1 shows an example of adaptive tetrahedral meshes extracted from a scanned CT data. As a byproduct, the uniform hexahedral mesh extraction algorithm is simpler.
We analyze each interior vertex a grid point inside the interval volume which is shared by eight cells. One minimizer is calculated for each of them, and those eight minimizers construct a hexahedron. Depending on the selected isovalues, different meshes of a skin and a skull are constructed.
The number of tetrahedra can be controlled by choosing a user specified error tolerance tet - b , c , d , e Note that the extracted mesh has no crack and no hanging node. Reconstructing a mesh with correct topology is important for accurate finite element calculations. The topology of the 3D mesh is preserved during the simplification process. Unlike the dual contouring method [ 31 ], we use a different error function based on the function difference normalized by gradients.
The function approximates the maximum difference between coarse and fine level isosurfaces to decide the level of adaptivity. Using this error measurement and a user specified error tolerance, we can identify octree cells of appropriate levels which satisfy the threshold criteria. The result shows the error function we use yields feature-sensitive adaptation as shown in Figure 13 d.
Since we still use QEF for computing minimizing vertices, we can also preserve sharp edges and corners. Tetrahedral meshes of a fandisk and b mechanical part.
Note that sharp edges and corners are accurately reconstructed; c , d: The facial features are better refined in d. The tetrahedral meshes extracted from volume data can not be used for finite element calculations directly, since some elements may have bad quality. The edge-ratio and Joe-Liu parameter are chosen to measure the mesh quality. The edge contraction method removes tetrahedra with bad edge-ratios, and the smoothing method improves the mesh quality measured by the Joe-Liu parameters.
We applied our algorithm to extracting a tetrahedral mesh from the accessibility function of a mouse acetylcholinesterase mAChE biomolecule. The extracted meshes have been used for efficient and correct finite element calculation of a diffusion problem [ 57 ].
The remainder of this paper is organized as the following: Section 2 summarizes the related work on quality 3D mesh generation; Section 3 reviews the preprocessing step. Section 7 discusses the mesh quality improvement. Section 8 shows some results and applications. The final section presents our conclusion. In most cases, an isosurface is extracted in the form of piecewise linear approximation for the modeling and rendering purpose.
The Marching Cubes algorithm MC [ 39 ] visits each cell in a volume and performs local triangulation based on the sign configuration of the eight vertices. To avoid visiting unnecessary cells, accelerated algorithms [ 64 ] [ 3 ] minimizing the time to search for contributing cells are developed.
The isosurfaces of a function defined by the trilinear interpolation inside a cubic cell may have a complicated shape and topology which cannot be reconstructed correctly using MC. The function values of face and body saddles in the cell can be used to decide the correct topology and consistent triangulation of an isosurface in the cell [ 42 ]. Lopes and Brodlie [ 38 ] provided a more accurate triangulation. Main drawbacks of MC and its variants are i the produced mesh is uniform, ii badly shaped triangles are generated.
An adaptive isosurface can be generated by triangulating cells with different levels. When the adjacent cubes have different resolution levels, the cracking problem will happen.
To keep the face compatibility, the gravity center of the coarser triangle is inserted, and a fan of triangles are used to approximate the isosurface [ 63 ]. The chain-gang algorithm [ 33 ] was presented for isosurface rendering of super adaptive resolution SAR and resolves discontinuities in SAR data sets.
Progressive multiresolution representation and recursive subdivision are combined effectively, and isosurfaces are constructed and smoothed by applying the edge bisection method [ 45 ]. A surface wave-front propagation technique [ 65 ] is used to generate multiresolution meshes with good aspect ratio.
The enhanced distance field representation and the extended MC algorithm [ 32 ] can detect and reconstruct sharp features in the isosurface. By combining SurfaceNets [ 28 ] and the extended Marching Cubes algorithm [ 32 ], octree based Dual Contouring [ 31 ] can generate adaptive isosurfaces with good aspect ratio and preservation of sharp features.
Elements in the extracted mesh often have bad aspect ratio. These elements can not be used for finite element calculations. The grid snapping method reduces the number of elements in an approximated isosurface and also improves the aspect ratio of the elements [ 41 ]. Octree based, advancing front based and Delaunay like techniques were used for tetrahedral mesh generation.
The octree technique recursively subdivides the cube containing the geometric model until the desired resolution is reached [ 51 ]. Advancing front methods start from a boundary and move a front from the boundary towards empty space within the domain [ 37 ] [ 23 ] [ 50 ]. Delaunay refinement is to refine the triangles or tetrahedra locally by inserting new nodes to maintain the Delaunay criterion. Different approaches to define new nodes were studied [ 16 ] [ 52 ] [ 14 ].
Sliver exudation [ 15 ] was used to eliminate those slivers. A deterministic algorithm [ 14 ] was presented for generating a weighted Delaunay mesh with no poor quality tetrahedra including slivers.
Shewchuk [ 53 ] solved the problem of enforcing boundary conformity by constrained Delaunay triangulation CDT. Delaunay refinement [ 52 ], edge removal and multi-face removal optimization algorithm [ 54 ] were used to improve the tetrahedral quality. Shewchuk [ 55 ] provided some valuable conclusions on quality measures for finite element method. MC was extended to extract tetrahedral meshes between two isosurfaces directly from volume data [ 24 ]. A Branch-on-Need octree was used as an auxiliary data structure to accelerate the extraction process.
A different algorithm, Marching Tetrahedra MT , was proposed for interval volume tetrahedralization [ 43 ]. A multiresolution framework [ 68 ] was generated by combining recursive subdivision and edge-bisection methods.
Since many 3D objects are sampled in terms of slices, Bajaj et. Eppstein [ 19 ] started from a tetrahedral mesh to decompose each tetrahedron into four hexahedra. Although this method avoids many difficulties, it increases the number of elements. There are four distinct methods for unstructured all-hex mesh generation: The grid-based approach generates a fitted 3D grid of hex elements on the interior of the volume [ 48 ] [ 49 ].
Medial surface methods involve an initial decomposition of the volume [ 46 ] [ 47 ]. Plastering places elements on boundaries first and advances towards the center of the volume [ 10 ] [ 8 ]. Whisker weaving first construct the spatial twist continuum STC or dual of the hex mesh, then the hex elements can be fitted into the volume using the STC as a guide [ 59 ].
Algorithms for mesh improvement can be classified into three categories [ 60 ] [ 44 ]: Laplacian smoothing, in its simplest form, relocates the vertex position at the average of the nodes connecting to it. This method generally works quite well for meshes in convex regions.
In the boundary cell, those faces with all four vertices lying inside the interval volume are called interior faces. Different from the boundary cell, all the eight vertices of an interior cell lie interior to the interval volume. For isosurface extraction, we only need to analyze boundary cells — those cells that contain sign change edges, or those cells that contain the isosurface.
There are four neighbor cubes which share the same sign change edge. Dual Contouring generates one minimal vertex for each neighbor cube by minimizing the QEF, then connects them to generate a quad. By marching all the sign change edges, the isosurface is obtained. For tetrahedral mesh extraction, cells inside the interval volume should also be analyzed besides the boundary cells.
Figure 3 is a uniform triangulation example of the area interior to the isocontour in two dimensions. There are three different cases which need to be dealt with separately. Uniform triangulation - the red curve represents the isocontour, green points represent minimizers. Compared to 2D triangulation, three dimensional tetrahedral meshing is more complicated.
Figure 4 shows the case table of uniform tetrahedralization. The case table of uniform tetrahedralization - the red vertex means it lies interior to the interval volume, otherwise, it is outside.
Green points represent minimizers. Our meshing algorithm assumes that there is only one minimizer point in a cell. This means two different boundary isosurfaces of an interval volume can not pass through the same cell. Therefore, we enforce every cell to have at most one boundary isosurface before actual meshing of cells.
If a cell contains two boundary isosurfaces, we recursively subdivide the cell into eight identical sub-cells until each sub-cell contains at most one boundary isosurface as shown in Figure 5.
The detailed subdivision algorithm is described in Section 5. A two dimensional example of cell subdivision for enforcing each cell to have at most one boundary isocontour. When two boundary isocontours pass through the same cell, the cell is recursively subdivided until each sub-cell contains at most one minimizer.
Uniform tetrahedralization usually generates an over-sampled mesh. Adaptive tetrahedral meshing is an effective way to minimize the number of elements while preserving the accuracy requirement. First, we split the volume data by using the octree data structure to obtain denser cells along the boundary, and coarser cells inside the interval volume Figure 7. The QEF value is calculated for each octree cell. We call this set of cells to be leaf cells of the octree assuming we pruned unnecessary nodes from the tree.
The red curve represents the isocontour, green points represent minimizers. The case table for decomposing the interior cell into triangles in 2D. The case table can be easily generalized to any other adaptive cases.
Each leaf cell may have neighbors at different levels. An edge in a leaf cell may be divided into several edges in its neighbor cells. Therefore it is important to decide which edge should be analyzed. The Dual Contouring method provides a good rule to follow — we always choose the minimal edge. Minimal edges are those edges of leaf cubes that do not properly contain an edge of a neighboring leaf.
Compared to the uniform case, the only difference is how to decompose interior cells into tetrahedra without hanging nodes. Generally, a hanging node is a point that is a vertex for some elements e. It lies on one edge or one face of its neighbors, for example, a T-Vertex. Figure 6 shows two methods to remove hanging nodes - splitting and merging. In the T-Vertex example Figure 6a , there is a hanging node red point. Only the right two triangles Number 2 and 3 need to be modified if we use the merging method Figure 6b ; while only the left one Number 1 needs to be modified if we intend to split the mesh Figure 6c.
In order to maintain the accuracy, we adopt the splitting method in our algorithm. Hanging node removal - the red point is a hanging node. The red curve is the real isocontour, and green points are minimizer points for boundary cells. Only the interior cell needs to be modified if the splitting method is adopted.
All the leaf cells can be divided into two groups: Figure 7 left shows an example of how to triangulate the interior area of an isocontour. Similarly, we need to analyze the following three problems:. Compared to the uniform case, the triangulation of interior cells is more complicated. All neighbors of an interior cell need to be checked because the neighbor cells are used to decide if there are any middle points on the shared edge.
So we need to search all the middle points on this edge by looking at the resolution levels of the neighbor cells. Figure 7 right lists all the main cases of how to decompose the interior cell into triangles according to its neighbors' resolution levels. If all the four edges have already been subdivided, then we can use the recursive method to march each of the four sub-cells with the same algorithm.
In this way, hanging nodes are removed effectively. For three dimensional adaptive tetrahedralization, we use the similar algorithm with the uniform case when we deal with the boundary cell. Any other adaptive cases are easily generalized using the case table in Figure 7. By using the above algorithm, we extract tetrahedral meshes from volumetric imaging data successfully. Figure 9 a and b show the tetrahedral mesh of the human head model extracted from 65 3 volumetric data. The volume inside the skin isosurface is tetrahedralized.
Finite element calculations sometimes require hexahedral meshes instead of tetrahedral meshes. Each hexahedron has eight points. In the tetrahedralization process, we deal with edges shared by at most four cells. This means that we can not get eight minimizers for each edge. However, each vertex within the interval volume is shared by eight cells. We can calculate a minimizer for each of them. In the case of interior cells, we set the center point as the minimizer.
These eight minimizers can then be used to construct a hexahedron. Figure 9 c and d show two hexahedral meshes for the head model, which are used to solve electromagnetic scattering simulations in finite element calculations. Constructing an adaptive 3D mesh with correct topology plays an important role in accurate and efficient finite element calculations.
Our goals in this section are i meshing with correct topology and ii topology preserving adaptive meshing. The topology of a mesh defined by an interval volume depends on the topology of two boundary isosurfaces enclosing the interval volume.
Therefore we focus on the topology of an isosurface. Assume the function F within a cubic cell is defined by the trilinear interpolation of the eight vertex values. The sign of a vertex is defined to be positive when its value is greater than or equal to an isovalue, and negative when its value is less than the isovalue.
There are 2 8 sign configurations which can be reduced into 14 cases using symmetry [ 39 ]. Several cases may have more than one local topology, and are termed ambiguous. We refer to [ 38 ] for all the cases of different local topology of an isosurface in a cube. When a case is ambiguous, the standard dual contouring method causes a non-manifold at the minimizing vertex in the cube. This makes the topology of the dual contour different from that of the real isosurface. On the other hand, if a cube has no ambiguity, then the real isosurface is topologically equivalent to the dual contour, which is always a simple manifold.
Whether a case in a cube is ambiguous or not can be checked by collapsing each edge which has two vertices with the same sign into a vertex [ 31 ]. If the cube can be collapsed into an edge, then the cube has no ambiguity and the topologically correct dual contour is generated in the cube. An example of a cell with an ambiguous case is shown in Figure 11 a where the non-manifold dual contour is generated using a naive approach.
An example of an ambiguous case for a finest-level cell. Both the front-right-up and back-left-down vertices have the positive sign, all the other vertices have the negative sign. A non-manifold dual contour is generated in a cube with an ambiguous case. In this case, the real isosurface in the cube is topologically equivalent to either two simple disks b or a tunnel c [ 38 ].
The topologically correct dual contours, b and c , can be constructed by the simple subdivision of a cell in a. A real example the human knee on topologically correct reconstruction of a dual contour. Note that non-manifolds in e are removed in f. We use the recursive cell subdivision in the finest level to reconstruct a dual contour with correct topology when the cell contains a non-manifold dual contour.
The subdivision algorithm is very similar with what we used for enforcing each cell to have at most one boundary isosurface. As a first step, all boundary cells in the finest level are identified. If a cell contains a non-manifold dual contour, we subdivide the cube into eight identical sub-cubes. The function values defined on newly generated vertices are calculated by the trilinear interpolation of the values at the eight cube vertices.
We recursively repeat this process for the sub-cubes containing a non-manifold dual contour until each sub-cube contains a manifold dual contour. Figure 10 shows a 2D example. A two dimensional example on the recursive subdivision of a cubic cell in the finest level for reconstructing a dual contour with correct topology.
We justify the correctness of the algorithm as follows. The function defined in a sub-cube is exactly same as the original function because we use the trilinear interpolation. If a sub-cube has no ambiguity and hence contains a manifold dual contour, then the topology of the dual contour in the sub-cube is correct in the sense that the dual contour is topologically equivalent to the real isosurface.
Therefore, if every sub-cube contains a manifold dual contour, then the dual contour in each sub-cube has correct topology. Since we recursively subdivide a cube until every sub-cube has a manifold dual contour, the resulting dual contour within the finest cubic cell has correct topology.
In this way, we can obtain a dual contour with correct topology from every finest cell in an octree structure. Then we traverse the octree in a bottom-up manner to get an adaptively simplified dual contour. During the traverse from children cells to a parent cell, the topology of a dual contour can change.
This may not be desirable. The paper [ 31 ] described an algorithm to check whether the fine dual contour is topologically equivalent to the coarse one or not. We restrict the octree simplification process to preserve the topology by using their algorithm. Figure 11 shows an example of a finest-level cell with an ambiguous case. A non-manifold dual contour is generated in the cell a.
However, the real isosurface in the cell can have either two disks b or a tunnel shape c d depending on an isovalue. The tunnel shape is correctly reconstructed as shown in d. Finite element applications require a minimal number of elements while preserving important features on boundary surfaces for efficient and accurate calculations. For a given precision requirement, uniform meshes are always over-sampled with unnecessary small elements.
Adaptive meshes are therefore preferable. The level adaptivity can be controlled manually by regions or automatically by using an error function and a tolerance. For example, in the calculation of ligand binding rate constants on mAChE data Figure 18 , the geometric accuracy on the cavity area mostly affects the accuracy of the calculation.
Therefore, we refine the cavity area as much as possible, while keep coarse meshes in other regions. For this purpose, we use an error function Equation 5 which approximates the difference between isosurfaces defined from two neighboring levels. Large geometric change of surfaces is considered as features. The error is measured as large in a region which contains important features, therefore the features are not easily lost during the process of adaptive simplification.
Similar error metric is used in [ 29 ]. The details of the difference approximation can be found in [ 58 ]. The color on the isosurface represents the distribution of the potential function, the color map is: Note the region around the cavity has fine meshes, while other areas have relatively coarse meshes. For level i , the eight red vertices' function values are given, and a trilinear function is defined in Equation 4 , from which the function values of 12 edge middle points green , 6 face middle points blue and 1 center point yellow can be obtained.
We want to estimate the difference of the isosurface between the two neighboring levels. In the right picture, the red curve represents the trilinear function in Level i it becomes to a straight line in 1D , and the green straight line represents the tangent line of the trilinear function at the middle point.
The error function is defined in Equation 5. The right picture of Figure 12 shows the calculation of the isosurface error U at the green middle point in 1D. In higher dimensions, the slope k becomes the magnitude of the gradient. In Equation 5 , we only need to sum the isosurface error over all the middle points, including edge middle points, face middle points and the center point, since the function values at the eight vertices are the same for the two neighboring levels.
We still use QEF to calculate minimizer points, which are connected to approximate the isosurfaces, so we can also preserve sharp edges and corners as shown in Figure 13 a and b.
QEF and the error function in Equation 5 are compared in controlling the level adaptivity for the human head model Figure 13, c and d , and Equation 5 yields more sensitive adaptivity for facial features, like the areas of nose, eyes, mouth and ears. The feature sensitive adaptivity is important for finite element meshes to identify and preserve necessary geometric and topological properties of the object while minimizing the number of elements. Poorly shaped elements influence the convergence and stability of the numerical solutions.
Since the extracted meshes may have such undesirable elements, we need an additional step for mesh quality improvement. First we need to define criteria to measure the mesh quality. Various functions have been chosen to measure the quality of a mesh element. For example, Freitag [ 22 ] defined poor quality tetrahedra using dihedral angles. George [ 26 ] chose the ratio of the element diameter such as the longest edge over the in-radius. We use the edge-ratio, the Joe-Liu parameter [ 35 ], and a minimum volume bound.
These quality metrics are used to detect slivers and sharp elements which need to be removed. With these measures, the mesh quality can be judged by observing the worst element quality, and the distribution of elements in terms of their quality values.
We perform iterative edge contractions to improve the worst edge-ratio. For each iteration, we remove the tetrahedron with the maximum edge-ratio by contracting the shortest edge.
We keep removing the tetrahedron with the maximum edge-ratio until the maximum edge-ratio is below the given threshold. During the edge contraction, we merge an interior vertex to a boundary vertex, an interior vertex to another interior vertex, or a boundary vertex to another boundary vertex.
A special case may occur when we contract edges as shown in Figure 14 , where two blue triangles left picture degenerate into the same triangle, and which should also be removed from the mesh. The special case can be detected by checking the number of elements sharing the same edge 2D or face 3D.
This can also be removed by contracting one shared edge. If edge contraction is not enough to arrive the threshold, then the longest edge bisection method is used to continue reducing the largest edge-ratio. But the number of vertices and the number of elements will increase. A special case for the edge-contraction method. Left - the original mesh, the red edge is to be contracted; Middle - the red edge is contracted; Right - the additional triangles are removed.
This process is executed when we extract meshes, therefore our mesh generation method tends to produce meshes with good overall quality. Finally smoothing techniques are used to improve the Joe-Liu parameter and the minimum volume. The simplest discretization of the Laplacian operator for a node is the average of all its neighbors.
Laplacian smoothing is an efficient heuristic, but it is possible to produce an invalid mesh containing inverted elements or elements with negative volume. The histogram of edge-ratios and Joe-Liu paramters Figure 15 shows the overall quality of extracted tetrahedral meshes for a biomolecule mAChE Figure 18 and the heart model Figure By comparing the two quality metrics before and after applying techniques of quality improvement, we can see the worst parameters are improved significantly.
Figure 16 shows the improvement of the worst values of the edge-ratio, the Joe-Liu parameter and the minimum volume. The number of tetrahedra at edge-ratio Heart SDF — a: The tetrahedral mesh extracted from SDF of the heart surface. The smooth surface and the wire frame on the mesh is rendered; c: Note that the region of heart valves are refined; d: Cross-section of the adaptive tetrahedral mesh. We have developed an interactive program for 3D mesh extraction and rendering from volume data.
In the program, error tolerances and isovalues can be changed interactively. Our algorithm has been used to generate tetrahedral meshes for a molecular dataset mAChE and the human heart model in two projects. We also tested our algorithm on volumetric data from CT scans, the UNC Human Head and Poly heart valve , and signed distance volumes generated from the polygonal surfaces of a human head and a knee.
The results consist of the number of tetrahedra, extraction time, and corresponding images with respect to different isovalues and error tolerances. Extraction time in the table includes octree traversal, QEF computation and actual mesh extraction, given isovalues and error tolerance values for inner and outer surfaces as run time parameters.
If we fix isovalues, and change error tolerance interactively, the computed QEF is reused and thus the whole extraction process is accelerated. The results show that the mesh extraction time scales linearly with the number of elements in the extracted mesh. Datasets and Test Results. The CT data sets are re-sampled to fit into the octree representation. Figure 18 shows the extracted adaptive tetrahedral mesh of mAChE, which has been used as the geometric model in solving the steady-state Smoluchowski equation to calculate ligand binding rate constants using the finite element method FEM [ 57 ].
The most important part in the geometric structure of mAChE is the cavity, where fine meshes are required. We first find the position of the cavity, then control the adaptivity according to it. The area of the cavity is kept the finest level, while coarser meshes are obtained everywhere else.
After improving the mesh quality, convergent results have been obtained in the finite element calculation, and they match with some experimental results well. A good geometric model of the human heart is important for the simulation of the human cardiovascular system, which is very useful for the predictive medicine applications in cardiovascular surgery.
To extract 3D meshes from the surface heart model provided by New York University, we computed the signed distance function from the surface data and performed the mesh extraction Figure An adaptive and quality tetrahedral mesh is extracted with correct topology and feature sensitive adaptation. Meshes are refined in the areas of heart valves, while coarse meshes are kept for other regions. The results from CT data are shown in Figure 1 skull, skin and 22 heart valve.
The number of elements in the extracted mesh is controlled by changing error tolerance. It is clear that adaptive tetrahedral meshes are extracted from the interval volume, and facial features are identified sensitively and preserved Figure In Figure 21 , the sequence of images are generated by changing the isovalue of the inner isosurface.
The topology of the inner isosurface can change arbitrarily. We have presented an algorithm to extract adaptive and high quality 3D meshes directly from volumetric imaging data. By extending the dual contouring method described in [ 31 ], our method can generate 3D meshes with good properties such as no hanging nodes, sharp feature preservation and good aspect ratio.
Using an error metric which is normalized by the function gradient, the resolution of the extracted mesh is adapted to the features sensitively. The resulting meshes are useful for efficient and accurate finite element calculations. National Center for Biotechnology Information , U. Comput Methods Appl Mech Eng. Author manuscript; available in PMC Sep Author information Copyright and License information Disclaimer. See other articles in PMC that cite the published article. Abstract This paper describes an algorithm to extract adaptive and quality 3D meshes directly from volumetric imaging data.
Open in a separate window. Tetrahedral Mesh Generation Octree based, advancing front based and Delaunay like techniques were used for tetrahedral mesh generation.
Hexahedral Mesh Generation Eppstein [ 19 ] started from a tetrahedral mesh to decompose each tetrahedron into four hexahedra. Quality Improvement Algorithms for mesh improvement can be classified into three categories [ 60 ] [ 44 ]: Here are definitions used in the algorithm description: Sign change edge — find the minimizers of the two cells which share the edge, then the two minimizers and the interior vertex of the edge construct a triangle blue triangles.
Interior edge in boundary cell — find the QEF minimizer of the boundary cell, then the minimizer and this interior edge construct a triangle yellow triangles. Sign change edge — decompose the quad into two triangles, then each triangle and the interior vertex of this edge construct a tetrahedron. In Figure 4 a , the red line represents the sign change edge, and two blue tetrahedra are constructed. Interior edge in boundary cell — find the QEF minimizers of the boundary cell and its boundary neighbor cells, then two adjacent minimizers and the interior edge construct a tetrahedron.
In Figure 4 b c , the red cube edge represents the interior edge. While c assumes the cell below this boundary cell is interior to the interval volume, so there is no minimizer for it. Therefore we obtain three minimizers, and only two tetrahedra are constructed. Interior face in boundary cell — find the QEF minimizer of the boundary cell, then the interior face and the minimizer construct a pyramid, which can be decomposed into two tetrahedra Figure 4 f.
Figure 4 d e f show a sequence of how to generate tetrahedra when there is only one interior face in the boundary cell.
Interior cell — decompose the interior cube into five tetrahedra. There are two different decomposition ways Figure 4 g h. For two adjacent cells, we choose a different decomposition method to avoid the diagonal choosing conflict problem. Interior cell — Since its neighbor cells may have higher resolution levels, hanging nodes are unavoidable. In Figure 6d , there is a hanging node if we triangulate the interior cell as in the uniform case.
Figure 6e shows a re-triangulating method to remove the hanging node. The two rules guarantee that no hanging nodes need to be removed for the boundary cell if only the splitting method is chosen. Similarly, we need to analyze the following three problems: Sign change edge — if the edge is minimal, deal with it as in the uniform case blue triangles. Interior edge in the boundary cell — if the edge is minimal, deal with it as in the uniform case yellow triangles.
Interior cell — Figure 7 Right lists all the main cases of how to decompose an interior cell into triangles. Sign change edge — if the edge is minimal, deal with it as in the uniform case.
Interior edge in the boundary cell — if the edge is minimal, deal with it as in the uniform case. Interior face in the boundary cell — identify all the middle points on the four edges, and decompose the face into triangles by applying the same algorithm as in the adaptive 2D case, then calculate the minimizer of this cell, each triangle and this minimizer construct a tetrahedron.
Typically, in mechanics , the prescribed exact solution consists of displacements that vary as piecewise linear functions in space called a constant strain solution. The elements pass the patch test if the finite element solution is the same as the exact solution.
It was long conjectured by engineers that passing the patch test is sufficient for the convergence of the finite element, that is, to ensure that the solutions from the finite element method converge to the exact solution of the partial differential equation as the finite element mesh is refined.
However, this is not the case, and the patch test is neither sufficient nor necessary for convergence. A broader definition of patch test applicable to any numerical method, including and beyond finite elements is any test problem having an exact solution that can, in principle, be exactly reproduced by the numerical approximation. Therefore, a finite-element simulation that uses linear shape functions has patch tests for which the exact solution must be piecewise linear, while higher-order finite elements have correspondingly higher-order patch tests.
From Wikipedia, the free encyclopedia. This article is about a numerical problem solving technique. For allergy testing, see Patch test. The Finite Element Method:
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