Shankar Prasad Sastry

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.

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.

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.

A parallel log-barrier method for mesh quality improvement and untangling

The development of parallel algorithms for mesh generation, untangling, and quality improvement is of high importance due to the need for large meshes with millions to billions of elements and the availability of supercomputers with hundreds to thousands of cores. There have been prior efforts in the development of parallel algorithms for mesh generation and local mesh quality improvement in which only one vertex is moved at a time. But for global mesh untangling and for global mesh quality improvement, where all vertices are simultaneously moved, parallel algorithms have not yet been developed. In our earlier work, we developed a serial global mesh optimization algorithm and used it to perform mesh untangling and mesh quality improvement. Our algorithm moved the vertices simultaneously to optimize a log-barrier objective function that was designed to untangle meshes as well as to improve the quality of the worst quality mesh elements. In this paper, we extend our work and develop a parallel log-barrier mesh untangling and mesh quality improvement algorithm for distributed-memory machines. We have used the algorithm with an edge coloring-based algorithm for synchronizing unstructured communication among the processes executing the log-barrier mesh optimization algorithm. The main contribution of this paper is a generic scheme for global mesh optimization, whereby the gradient of the objective function with respect to the position of some of the vertices is communicated among all processes in every iteration. The algorithm was implemented using the OpenMPI 2.0 parallel programming constructs and shows greater strong scaling efficiency compared to an existing parallel mesh quality improvement technique.

Performance characterization of nonlinear optimization methods for mesh quality improvement

We characterize the performance of gradient- and Hessian-based optimization methods for mesh quality improvement. In particular, we consider the steepest descent and Polack-Ribière conjugate gradient methods which are gradient based. In the Hessian-based category, we consider the quasi-Newton, trust region, and feasible Newton methods. These techniques are used to improve the quality of a mesh by repositioning the vertices, where the overall mesh quality is measured by the sum of the squares of individual elements according to the aspect ratio metric. The effects of the desired degree of accuracy in the improved mesh, problem size, initial mesh configuration, and heterogeneity in element volume on the performance of the optimization solvers are characterized on a series of tetrahedral meshes.

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 Comparison of Gradient- and Hessian-Based Optimization Methods for Tetrahedral Mesh Quality Improvement

Discretization methods, such as the finite element method, are commonly used in the solution of partial differential equations (PDEs). The accuracy of the computed solution to the PDE depends on the degree of the approximation scheme, the number of elements in the mesh [1], and the quality of the mesh [2, 3]. More specifically, it is known that as the element dihedral angles become too large, the discretization error in the finite element solution increases [4]. In addition, the stability and convergence of the finite element method is affected by poor quality elements. It is known that as the angles become too small, the condition number of the element matrix increases [5].

Adaptive and Unstructured Mesh Cleaving.

We propose a new strategy for boundary conforming meshing that decouples the problem of building tetrahedra of proper size and shape from the problem of conforming to complex, non-manifold boundaries. This approach is motivated by the observation that while several methods exist for adaptive tetrahedral meshing, they typically have difficulty at geometric boundaries. The proposed strategy avoids this conflict by extracting the boundary conforming constraint into a secondary step. We first build a background mesh having a desired set of tetrahedral properties, and then use a generalized stenciling method to divide, or “cleave”, these elements to get a set of conforming tetrahedra, while limiting the impacts cleaving has on element quality. In developing this new framework, we make several technical contributions including a new method for building graded tetrahedral meshes as well as a generalization of the isosurface stuffing and lattice cleaving algorithms to unstructured background meshes.

On Interpolation Errors over Quadratic Nodal Triangular Finite Elements

Interpolation techniques are used to estimate function values and their derivatives at those points for which a numerical solution of any equation is not explicitly evaluated. In particular, the shape functions are used to interpolate a solution (within an element) of a partial differential equation obtained by the finite element method. Mesh generation and quality improvement are often driven by the objective of minimizing the bounds on the error of the interpolated solution. For linear elements, the error bounds at a point have been derived as a composite function of the shape function values at the point and its distance from the element’s nodes. We extend the derivation to quadratic triangular elements and visualize the bounds for both the function interpolant and the interpolant of its derivative. The maximum error bound for a function interpolant within an element is computed using the active set method for constrained optimization. For the interpolant of the derivative, we visually observe that the evaluation of the bound at the corner vertices is sufficient to find the maximum bound within an element.We characterize the bounds and develop a mesh quality improvement algorithm that optimizes the bounds through the movement (r-refinement) of both the corner vertices and edge nodes in a high-order mesh.