Jibum Kim

A numerical investigation on the interplay amongst geometry, meshes, and linear algebra in the finite element solution of elliptic PDEs

In this paper, we study the effect of the choice of mesh quality metric, preconditioner, and sparse linear solver on the numerical solution of elliptic partial differential equations (PDEs). We smooth meshes on several geometric domains using various quality metrics and solve the associated elliptic PDEs using the finite element method. The resulting linear systems are solved using various combinations of preconditioners and sparse linear solvers. We use the inverse mean ratio and radius ratio metrics in addition to conditioning-based scale-invariant and interpolation-based size-and-shape metrics. We employ the Jacobi, SSOR, incomplete LU, and algebraic multigrid preconditioners and the conjugate gradient, minimum residual, generalized minimum residual, and bi-conjugate gradient stabilized solvers. We focus on determining the most efficient quality metric, preconditioner, and linear solver combination for the numerical solution of various elliptic PDEs with isotropic coefficients. We also investigate the effect of vertex perturbation and the effect of increasing the problem size on the number of iterations required to converge and on the solver time. In this paper, we consider Poisson’s equation, general second-order elliptic PDEs, and linear elasticity problems.

Patient-Specific Model Generation and Simulation for Pre-operative Surgical Guidance for Pulmonary Embolism Treatment

Pulmonary embolism (PE) is a potentially-fatal disease in which blood clots (i.e., emboli) break free from the deep veins in the body and migrate to the lungs. In order to prevent PE, anticoagulation therapy is often used; however, for some patients, it is contraindicated. For such patients, a mechanical filter, namely an inferior vena cava (IVC) filter, is inserted into the IVC to capture and prevent emboli from reaching the lungs. There are numerous IVC filter designs, and it is not well understood which particular IVC filter geometry will result in the best clinical outcome for a given patient. Patient-specific computational fluid dynamic (CFD) simulations may be used to aid physicians in IVC filter selection and placement. In particular, such computational simulations may be used to determine the capability of various IVC filters in various positions to capture emboli, while not creating additional emboli or significantly altering the flow of blood in the IVC. In this paper, we propose a computational pipeline that can be used to generate patient-specific geometric models and computational meshes of the IVC and IVC filter for various IVC anatomies based on the patient’s computer tomography (CT) images. Our pipeline involves several steps including image processing, geometric model construction, surface and volume mesh generation, and CFD simulation. We then use our patient-specific meshes of the IVC and IVC filter in CFD simulations of blood flow, whereby we demonstrate the potential utility of this approach for optimized, patient-specific IVC filter selection and placement for improved prevention of PE. The novelty of our approach lies in the use of a superelastic mesh warping technique to virtually place the surface mesh of the IVC filter (which was created via computer-aided design modeling) inside the surface mesh of the patient-specific IVC, reconstructed from clinical CT data. We also employ a linear elastic mesh warping technique to simulate the deformation of the IVC when the IVC filter is placed inside of it.

A Derivative-Free Mesh Optimization Algorithm for Mesh Quality Improvement and Untangling

We propose a derivative-free mesh optimization algorithm, which focuses on improving the worst element quality on the mesh. The mesh optimization problem is formulated as a min-max problem and solved by using a downhill simplex (amoeba) method, which computes only a function value without needing a derivative of Hessian of the objective function. Numerical results show that the proposed mesh optimization algorithm outperforms the existing mesh optimization algorithm in terms of improving the worst element quality and eliminating inverted elements on the mesh.

Efficient Solution of Elliptic Partial Differential Equations via Effective Combination of Mesh Quality Metrics, Preconditioners, and Sparse Linear Solvers

In this paper, we study the effect the choice of mesh quality metric, preconditioner, and sparse linear solver have on the numerical solution of elliptic partial differential equations (PDEs). We smoothe meshes on several geometric domains using various quality metrics and solve the associated elliptic PDEs using the finite element method. The resulting linear systems are solved using various combinations of preconditioners and sparse linear solvers. We use the inverse mean ratio and vertex condition number metrics in addition to interpolation-based, scale-variant and scale-invariant metrics. We employ the Jacobi, incomplete LU, and SSOR preconditioners and the conjugate gradient, minimum residual, generalized minimum residual, and bi-conjugate gradient stabilized solvers. We focus on determining the most efficient quality metric/preconditioner/linear solver combination for the numerical solution of various elliptic PDEs.

A Multiobjective Mesh Optimization Algorithm for Improving the Solution Accuracy of PDE Computations

Mesh qualities affect both the efficiency and accuracy for solving partial differential equations (PDEs). In this paper, we present a multiobjective mesh optimization algorithm, which improves the accuracy for solving PDEs. Our algorithm is designed to simultaneously improve more than two aspects of the mesh, while being able to successfully decrease errors for solving various PDEs. Numerical experiments show that our algorithm is able to significantly decrease errors compared with existing single objective mesh optimization algorithms.

A 2D Topology-Adaptive Mesh Deformation Framework for Mesh Warping

We propose a framework for performing anisotropic mesh deformations. Our goal is to produce high quality meshes with no inverted elements on domains which undergo large deformations. To the greatest extent possible, the meshes should have similar element shape; however, topological changes are performed as necessary in order to improve mesh quality. Our framework is based upon the previous work of two of the authors and their collaborators (Kim et al., Int. J. Numer. Methods Eng. 94(1):20–42, 2013; Kim et al., Computer and Mathematics with Applications, Submitted, November 2014) and consists of four steps. The first step is to perform anisotropic finite element-based mesh warping to estimate the interior vertex positions based upon an appropriate choice of the PDE coefficients. The second step is to perform multiobjective mesh optimization in order to eliminate inverted elements and improve element shape. Edge swaps are then performed to further improve the mesh quality. A final mesh smoothing pass is then performed. Our numerical results show that our framework can be used to generate high quality meshes with no inverted elements for very large deformations. In particular, the addition of topological changes to our hybrid mesh deformation algorithm (Kim et al., Computer and Mathematics with Applications, Submitted, November 2014) proved to be an extremely efficient way of improving the mesh quality.

A Hybrid Algorithm for Online Location Update using Feature Point Detection for Portable Devices

We propose a cost-efficient hybrid algorithm for online location updates that efficiently combines feature point detection with the online trajectory-based sampling algorithm. Our algorithm is designed to minimize the average trajectory error with the minimal number of sample points. The algorithm is composed of 3 steps. First, we choose corner points from the map as sample points because they will most likely cause fewer trajectory errors. By employing the online trajectory sampling algorithm as the second step, our algorithm detects several missing and important sample points to prevent unwanted trajectory errors. The final step improves cost efficiency by eliminating redundant sample points on straight paths. We evaluate the proposed algorithm with real GPS trajectory data for various bus routes and compare our algorithm with the existing one. Simulation results show that our algorithm decreases the average trajectory error 28% compared to the existing one. In terms of cost efficiency, simulation results show that our algorithm is 29% more cost efficient than the existing one with real GPS trajectory data.

Segment Delay Learning From Quantized Path Delay Measurements

Our understanding on a silicon chip is limited due to low measurement resolution or model-silicon miscorrelation including variations. This paper shows that chips are better understood by combining noisy measurement results and model information through a mathematical algorithm. Our proposed method learns segment delays in logic circuits from quantized path delay measurements using ridge regression. During the learning process, we take advantage of both nominal segment delays and the delay sensitivity with respect to variations. We also interpret the ridge regression in Bayesian context and in doing so, propose an analytic formula to set the regularization parameter of the ridge regression. For the silicon measurement environments where low measurement resolution is the dominant source of measurement noise, this formula allows us to predict post-silicon results more accurately and speed up the algorithm eliminating inefficient and inaccurate cross-validation. We also demonstrate our method in enhancing the resolution of already measured path delays. We learn segment delays from quantized path delay measurements and predict the path delays prior to the quantization. Our simulation results show that the predicted path delays are much closer to actual values than the measured values and the nominal values.

Untangling polygonal and polyhedral meshes via mesh optimization

We propose simple and efficient optimization-based untangling strategies for 2D polygonal and 3D polyhedral meshes. The first approach uses a size-based mesh metric, which eliminates inverted elements by averaging element size over the entire mesh. The second method uses a hybrid quality metric, which untangles inverted elements by simultaneously averaging element size and improving element shape. The last method using a variant of the hybrid quality metric gives a high penalty for inverted elements and employs an adaptive sigmoid function for handling various mesh sizes. Numerical experiments are presented to show the effectiveness of the proposed untangling strategies for various 2D polygonal and 3D polyhedral meshes.

An Iterative Mesh Untangling Algorithm Using Edge Flip

Existing mesh untangling algorithms are unable to untangle highly tangled meshes. In this study, we address this problem by proposing an iterative mesh untangling algorithm using edge flip. Our goal is to produce meshes with no inverted elements and good element qualities when inverted elements with poor element qualities are produced during mesh generation or mesh deformation process. Our proposed algorithm is composed of three steps: first, we iteratively perform edge flip; subsequently, optimization-based mesh untangling is conducted until all inverted elements are eliminated; finally, we perform mesh smoothing for generating high-quality meshes. Numerical results show that the proposed algorithm is able to successfully generate high-quality meshes with no inverted elements for highly tangled meshes.