Suzanne M. Shontz

A comparison of two optimization methods for mesh quality improvement

We compare inexact Newton and block coordinate descent optimization methods for improving the quality of a mesh by repositioning the vertices, where the overall quality is measured by the harmonic mean of the mean-ratio metric. The effects of problem size, element size heterogeneity, and various vertex displacement schemes on the performance of these algorithms are assessed for a series of tetrahedral meshes.

A Comparison of Mesh Morphing Methods for 3D Shape Optimization

The ability to automatically morph an existing mesh to conform to geometry modifications is a necessary capability to enable rapid prototyping of design variations. This paper compares six methods for morphing hexahedral and tetrahedral meshes, including the previously published FEMWARP and LBWARP methods as well as four new methods. Element quality and performance results show that different methods are superior on different models. We recommend that designers of applications that use mesh morphing consider both the FEMWARP and a linear simplex based method.

Analysis of and workarounds for element reversal for a finite element-based algorithm for warping triangular and tetrahedral meshes

We consider an algorithm called FEMWARP for warping triangular and tetrahedral finite element meshes that computes the warping using the finite element method itself. The algorithm takes as input a two- or three-dimensional domain defined by a boundary mesh (segments in one dimension or triangles in two dimensions) that has a volume mesh (triangles in two dimensions or tetrahedra in three dimensions) in its interior. It also takes as input a prescribed movement of the boundary mesh. It computes as output updated positions of the vertices of the volume mesh. The first step of the algorithm is to determine from the initial mesh a set of local weights for each interior vertex that describes each interior vertex in terms of the positions of its neighbors. These weights are computed using a finite element stiffness matrix. After a boundary transformation is applied, a linear system of equations based upon the weights is solved to determine the final positions of the interior vertices.The FEMWARP algorithm has been considered in the previous literature (e.g., in a 2001 paper by Baker). FEMWARP has been successful in computing deformed meshes for certain applications. However, sometimes FEMWARP reverses elements; this is our main concern in this paper. We analyze the causes for this undesirable behavior and propose several techniques to make the method more robust against reversals. The most successful of the proposed methods includes combining FEMWARP with an optimization-based untangler.

A log-barrier method for mesh quality improvement and untangling

The presence of a few inverted or poor-quality mesh elements can negatively affect the stability, convergence and efficiency of a finite element solver and the accuracy of the associated partial differential equation solution. We propose a mesh quality improvement and untangling method that untangles a mesh with inverted elements and improves its quality. Worst element mesh quality improvement and untangling can be formulated as a nonsmooth unconstrained optimization problem, which can be reformulated as a smooth constrained optimization problem. Our technique solves the latter problem using a log-barrier interior point method and uses the gradient of the objective function to efficiently converge to a stationary point. The method uses a logarithmic barrier function and performs global mesh quality improvement. We have also developed a smooth quality metric that takes both signed area and the shape of an element into account. This quality metric assigns a negative value to an inverted element. It is used with our algorithm to untangle a mesh by improving the quality of an inverted element to a positive value. Our method usually yields better quality meshes than existing methods for improvement of the worst quality elements, such as the active set, pattern search, and multidirectional search mesh quality improvement methods. Our method is faster and more robust than existing methods for mesh untangling, such as the iterative stiffening method.

Towards high-quality, untangled meshes via a force-directed graph embedding approach

Abstract High quality meshes are crucial for the solution of partial differential equations (PDEs) via the finite element method (or other PDE solvers). The accuracy of the PDE solution, and the stability and conditioning of the stiffness matrix depend upon the mesh quality. In addition, the mesh must be untangled in order for the finite element method to generate physically valid solutions. Tangled meshes, i.e., those with inverted mesh elements, are sometimes generated via large mesh deformations or in the mesh generation process. Traditional techniques for untangling such meshes are based on geometry and/or optimization. Optimization-based mesh untangling techniques first untangle the mesh and then smoothe the resulting untangled mesh in order to obtain high quality meshes; such techniques require the solution of two optimization problems. In this paper, we study how to modify a physical, force-directed method based upon the Fruchterman-Reingold (FR) graph layout algorithm so that it can be used for untangling. The objectives of aesthetic graph layout, such as minimization of edge intersections and near equalization of edge lengths, follow the goals of mesh untangling and generating good quality elements, respectively. Therefore, by using the force-directed method, we can achieve both steps of mesh untangling and mesh smoothing in one operation. We compare the effectiveness of our method with that of the optimization-based mesh untangling method in [1] and implemented in Mesquite by untangling a suite of unstructured triangular, quadrilateral, and tetrahedral finite element volume meshes. The results show that the force-directed method is substantially faster than the Mesquite mesh untangling method without sacrificing much in terms of mesh quality for the majority of the test cases we consider in this paper. The force-directed mesh untangling method demonstrates the most promise on convex geometric domains. Further modifications will be made to the method to improve its ability to untangle meshes on non-convex domains.

A Log-Barrier Method for Mesh Quality Improvement

The presence of a few poor-quality mesh elements can negatively affect the stability and efficiency of a finite element solver and the accuracy of the associated partial differential equation solution. We propose a mesh quality improvement method that improves the quality of the worst elements. Mesh quality improvement of the worst elements can be formulated as a nonsmooth unconstrained optimization problem, which can be reformulated as a smooth constrained optimization problem. Our technique solves the latter problem using a log-barrier interior point method and uses the gradient of the objective function to efficiently converge to a stationary point. The technique can be used with convex or nonconvex quality metrics. The method uses a logarithmic barrier function and performs global mesh quality improvement. Our method usually yields better quality meshes than existing methods for improvement of the worst quality elements, such as the active set, pattern search, and multidirectional search mesh quality improvement methods.

Two Derivative-Free Optimization Algorithms for Mesh Quality Improvement

High-quality meshes are essential in the solution of partial differential equations (PDEs), which arise in numerous science and engineering applications, as the mesh quality affects the solution accuracy, the solver execution time, and the problem conditioning. Mesh quality improvement is necessary when the mesh is of less than desirable quality (either from mesh generation or deformation). Nondifferentiable objective functions arise when the goal of the mesh optimization is to improve the worst quality element in the mesh. We propose two derivative-free methods for mesh optimization, namely the pattern search (PS) and multidirectional search (MDS) mesh quality improvement methods, to be used with nondifferentiable objective functions representing the overall mesh quality. Experimental results show that these two methods are successful in improving the worst quality mesh elements. The PS method yielded higher quality 2D meshes than did the MDS method; however, its execution time was longer. In the 3D case, most of the meshes converged to meshes of approximately the same quality because the initial meshes were fairly close to optimal. In 3D, the PS method required longer to execute than did the MDS method.

CPU-GPU Algorithms for Triangular Surface Mesh Simplification

Mesh simplification and mesh compression are important processes in computer graphics and scientific computing, as such contexts allow for a mesh which takes up less memory than the original mesh. Current simplification and compression algorithms do not take advantage of both the central processing unit (CPU) and the graphics processing unit (GPU). We propose three simplification algorithms, one of which runs on the CPU and two of which run on the GPU. We combine these algorithms into two CPU-GPU algorithms for mesh simplification. Our CPU-GPU algorithms are the naive marking algorithm and the inverse reduction algorithm. Experimental results show that when the algorithms take advantage of both the CPU and the GPU, there is a decrease in running time for simplification compared to performing all of the computation on the CPU. The marking algorithm provides higher simplification rates than the inverse reduction algorithm, whereas the inverse reduction algorithm has a lower running time than the marking algorithm.

A robust solution procedure for hyperelastic solids with large boundary deformation

Compressible Mooney–Rivlin theory has been used to model hyperelastic solids, such as rubber and porous polymers, and more recently for the modeling of soft tissues for biomedical tissues, undergoing large elastic deformations. We propose a solution procedure for Lagrangian finite element discretization of a static nonlinear compressible Mooney–Rivlin hyperelastic solid. We consider the case in which the boundary condition is a large prescribed deformation, so that mesh tangling becomes an obstacle for straightforward algorithms. Our solution procedure involves a largely geometric procedure to untangle the mesh: solution of a sequence of linear systems to obtain initial guesses for interior nodal positions for which no element is inverted. After the mesh is untangled, we take Newton iterations to converge to a mechanical equilibrium. The Newton iterations are safeguarded by a line search similar to one used in optimization. Our computational results indicate that the algorithm is up to 70 times faster than a straightforward Newton continuation procedure and is also more robust (i.e., able to tolerate much larger deformations). For a few extremely large deformations, the deformed mesh could only be computed through the use of an expensive Newton continuation method while using a tight convergence tolerance and taking very small steps.

A Computational Method for Predicting Inferior Vena Cava Filter Performance on a Patient-Specific Basis

A computational methodology for simulating virtual inferior vena cava (IVC) filter placement and IVC hemodynamics was developed and demonstrated in two patient-specific IVC geometries: a left-sided IVC and an IVC with a retroaortic left renal vein. An inverse analysis was performed to obtain the approximate in vivo stress state for each patient vein using nonlinear finite element analysis (FEA). Contact modeling was then used to simulate IVC filter placement. Contact area, contact normal force, and maximum vein displacements were higher in the retroaortic IVC than in the left-sided IVC (144 mm(2), 0.47 N, and 1.49 mm versus 68 mm(2), 0.22 N, and 1.01 mm, respectively). Hemodynamics were simulated using computational fluid dynamics (CFD), with four cases for each patient-specific vein: (1) IVC only, (2) IVC with a placed filter, (3) IVC with a placed filter and model embolus, all at resting flow conditions, and (4) IVC with a placed filter and model embolus at exercise flow conditions. Significant hemodynamic differences were observed between the two patient IVCs, with the development of a right-sided jet, larger flow recirculation regions, and lower maximum flow velocities in the left-sided IVC. These results support further investigation of IVC filter placement and hemodynamics on a patient-specific basis.