An Alternating Mesh Quality Metric Scheme for Efficient Mesh Quality Improvement

AbstractIn the numerical solution of partial differential equations (PDEs), high-quality meshes are crucial for the stability, accuracy, and convergence of the associated PDE solver. Mesh quality improvement is often performed to improve the quality of meshes before use in numerical solution of the PDE. Mesh smoothing (performed via optimization) is one popular technique for improving the mesh quality; it does so by making adjustments to the vertex locations. When an inefficient mesh quality metric is used to design the optimization problem, and hence also to measure the mesh quality within the optimization procedure, convergence of the optimization method can be much slower than desired. However, for many applications, the choice of mesh quality metric and the optimization problem should be considered fixed. In this paper, we propose a simple mesh quality metric alternation scheme for use in the mesh optimization process. The idea is to alternate the use of the original inefficient mesh quality metric with a more efficient mesh quality metric on alternate iterations of the mesh optimization procedure in order to reduce the time to convergence, while solving the original mesh quality improvement problem. Typical results of using our application scheme to solve mesh quality improvement problems yield approximately 40-55% improvement in comparison to the original mesh optimization procedure. More frequent use of the efficient metric results in greater speed-ups

Numerical Methods for Electronic Structure Calculations of Materials

This is the published version. Copyright 2010 Society for Industrial and Applied MathematicsThe goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems. The paper is intended for a diverse scientific computing audience. For this reason, we assume the reader does not have an extensive background in the related physics. Our overview focuses on the nature of the numerical problems to be solved, their origin, and the methods used to solve the resulting linear algebra or nonlinear optimization problems. It is common knowledge that the behavior of matter at the nanoscale is, in principle, entirely determined by the Schrödinger equation. In practice, this equation in its original form is not tractable. Successful but approximate versions of this equation, which allow one to study nontrivial systems, took about five or six decades to develop. In particular, the last two decades saw a flurry of activity in developing effective software. One of the main practical variants of the Schrödinger equation is based on what is referred to as density functional theory (DFT). The combination of DFT with pseudopotentials allows one to obtain in an efficient way the ground state configuration for many materials. This article will emphasize pseudopotential-density functional theory, but other techniques will be discussed as well

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.Comment: Revision of earlier version of paper. Submitted for publication in Engineering with Computers on 9 September 2010. Accepted for publication on 20 May 2011. Published online 11 June 2011. The final publication is available at http://www.springerlink.co

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 succesful 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.Comment: Revision of earlier version of paper. Submitted for publication in BIT Numerical Mathematics on 27 April 2010. Accepted for publication on 7 September 2010. Published online on 9 October 2010. The final publication is available at http://www.springerlink.co

Research and Education in Computational Science and Engineering

Over the past two decades the field of computational science and engineering (CSE) has penetrated both basic and applied research in academia, industry, and laboratories to advance discovery, optimize systems, support decision-makers, and educate the scientific and engineering workforce. Informed by centuries of theory and experiment, CSE performs computational experiments to answer questions that neither theory nor experiment alone is equipped to answer. CSE provides scientists and engineers of all persuasions with algorithmic inventions and software systems that transcend disciplines and scales. Carried on a wave of digital technology, CSE brings the power of parallelism to bear on troves of data. Mathematics-based advanced computing has become a prevalent means of discovery and innovation in essentially all areas of science, engineering, technology, and society; and the CSE community is at the core of this transformation. However, a combination of disruptive developments---including the architectural complexity of extreme-scale computing, the data revolution that engulfs the planet, and the specialization required to follow the applications to new frontiers---is redefining the scope and reach of the CSE endeavor. This report describes the rapid expansion of CSE and the challenges to sustaining its bold advances. The report also presents strategies and directions for CSE research and education for the next decade.Comment: Major revision, to appear in SIAM Revie

Prediction of ventricular boundary evolutionin hydrocephalic brain via a combined level set and adaptive finite element mesh warping method

Hydrocephalus is a serious neurological disorder caused by abnormalities in cerebrospinal fluid (CSF) flow, resulting in large brain deformations and neuronal damage. Treatments drain the excess CSF from the ventricles either via a CSF shunt or an endoscopic third ventriculostomy. However, patients’responseto these treatments is poor. Mathematical models of hydrocephalus mechanics couldaid neurosurgeons in hydrocephalus treatment planning.Current models and corresponding computational simulations of hydrocephalus are still in their infancy, despitethis being a disease with serious long-term implications.We propose a novel geometriccomputational approach for tracking the evolution of hydrocephalic brain tissue –ventricular CSF interfacevia the level set method and an adaptive mesh warping technique. In our previous work [1], weevolved the ventricular boundary in 2D CT images which required a backtracking line search for obtaining valid intermediate meshes for use with FEMWARP, a finite element-based mesh warping method. In [2], we automatically evolved the ventricular boundary deformation for 2D CT imagesvia the level set method. To help surgeons determine where to implant a shunt, we also computed the brain ventricle volume evolution for 3D MR images using FEMWARP.In this work, we generalize the results from [1,2] and incorporate adaptive mesh refinement following mesh deformation. Based on the use of the solution gradientof the PDE in the mesh warping approach as an enrichment indicator, the mesh is refinedwhere the solution gradient islargeand is coarsened where it is small.We present computational simulations of the onset and treatment of pediatric hydrocephalus based on 2D CT images which demonstrate the success of our combined level set/adaptive finite element-based mesh warping approach

A Mesh Warping Algorithm Based on Weighted Laplacian Smoothing

We present a new mesh warping algorithm for tetrahedral meshes based upon weighted laplacian smoothing. We start with a 3D domain which is bounded by a triangulated surface mesh and has a tetrahedral volume mesh as its interior. We then suppose that a movement of the surface mesh is prescribed and use our mesh warping algorithm to update the nodes of the volume mesh. Our method determines a set of local weights for each interior node which describe the relative distances of the node to its neighbors. After a boundary transformation is applied, the method solves a system of linear equations based upon the weights to determine the final position of the interior nodes. We study mesh invertibility and prove a theorem which gives su#cient conditions for a mesh to resist inversion by a transformation. We prove that our algorithm yields exact results for a#ne mappings and state a conjecture for more general mappings. In addition, we prove that our algorithm converges to the same point as both the local weighted laplacian smoothing algorithm and the Gauss-Seidel algorithm for linear systems. We test our algorithm's robustness and present some numerical results. Finally, we use our algorithm to study the movement of the canine heart