Mark S. Shephard

Automatic three-dimensional mesh generation by the finite octree technique

An octree-based fully automatic three-dimensional mesh generator is presented. The mesh generator is capable of meshing non-manifold models of arbitrary geometric complexity through the explicit tracking and enforcement of geometric compatibility and geometric similarity at each step of the meshing process. The resulting procedure demonstrates a linear growth rate with respect to the number of elements and can be easily integrated with any geometric modeller through a set of geometric operators.

Computational plasticity for composite structures based on mathematical homogenization: Theory and practice

This paper generalizes the classical mathematical homogenization theory for heterogeneous medium to account for eigenstrains. Starting from the double scale asymptotic expansion for the displacement and eigenstrain fields we derive a close form expression relating arbitrary eigenstrains to the mechanical fields in the phases. The overall structural response is computed using an averaging scheme by which phase concentration factors are computed in the average sense for each micro-constituent, and history data is updated at two points (reinforcement and matrix) in the microstructure, one for each phase. Macroscopic history data is stored in the database and then subjected in the post-processing stage onto the unit cell in the critical locations. For numerical examples considered, the CPU time obtained by means of the two-point averaging scheme with variational micro-history recovery with 30 seconds on SPARC 1051 as opposed to 7 hours using classical mathematical homogenization theory. At the same time the maximum error in the microstress field in the critical unit cell was only 3.5% in comparison with the classical mathematical homogenization theory.

A Modified Quadtree Approach To Finite Element Mesh Generation

By allowing the use of quadrants with cut corners, this modeling technique overcomes some of the drawbacks of standard quadtree encoding for finite element mesh generation.

Adaptive Local Refinement with Octree Load Balancing for the Parallel Solution of Three-Dimensional Conservation Laws

Conservation laws are solved by a local Galerkin finite element procedure with adaptive space-time mesh refinement and explicit time integration. The Courant stability condition is used to select smaller time steps on smaller elements of the mesh, thereby greatly increasing efficiency relative to methods having a single global time step. Processor load imbalances, introduced at adaptive enrichment steps, are corrected by using traversals of an octree representing a spatial decomposition of the domain. To accommodate the variable time steps, octree partitioning is extended to use weights derived from element size. Partition boundary smoothing reduces the communications volume of partitioning procedures for a modest cost. Computational results comparing parallel octree and inertial partitioning procedures are presented for the three-dimensional Euler equations of compressible flow solved on an IBM SP2 computer.

A GENERAL TOPOLOGY-BASED MESH DATA STRUCTURE

SUMMARY A representation for a mesh based on the topological hierarchy of vertices, edges, faces and regions, is described. The representation is general and easily supports procedures ranging from mesh generation to adaptive analysis processes. Three implementations are given which concentrate on di⁄erent aspects of performance (storage requirements and speed). Comparisons are made to other published representations. ( 1997 by John Wiley & Sons, Ltd.

An Adaptive Discontinuous Galerkin Technique with an Orthogonal Basis Applied to Compressible Flow Problems

We present a high-order formulation for solving hyperbolic conservation laws using the discontinuous Galerkin method (DGM). We introduce an orthogonal basis for the spatial discretization and use explicit Runge--Kutta time discretization. Some results of higher order adaptive refinement calculations are presented for inviscid Rayleigh--Taylor flow instability and shock reflection problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities.

Multiphysics simulations: Challenges and opportunities

We consider multiphysics applications from algorithmic and architectural perspectives, where “algorithmic” includes both mathematical analysis and computational complexity, and “architectural” includes both software and hardware environments. Many diverse multiphysics applications can be reduced, en route to their computational simulation, to a common algebraic coupling paradigm. Mathematical analysis of multiphysics coupling in this form is not always practical for realistic applications, but model problems representative of applications discussed herein can provide insight. A variety of software frameworks for multiphysics applications have been constructed and refined within disciplinary communities and executed on leading-edge computer systems. We examine several of these, expose some commonalities among them, and attempt to extrapolate best practices to future systems. From our study, we summarize challenges and forecast opportunities.

PARALLEL REFINEMENT AND COARSENING OF TETRAHEDRAL MESHES

This paper presents a parallel adaptation procedure (coarsening and refinement) for tetrahedral meshes in a distributed environment. Coarsening relies upon an edge collapsing tool. Refinement uses edge-based subdivision templates. Mesh optimization maintains the quality of the adapted meshes. Focus is given to the parallelization of the various components. Scalability requires repartitioning of the mesh before applying either coarsening or refinement. Relatively good speed-ups have been obtained for all phases of the proposed adaptation scheme. Copyright

Boundary layer mesh generation for viscous flow simulations

Viscous flow problems exhibit boundary layers and free shear layers in which the solution gradients, normal and tangential to the flow, differ by orders of magnitude. The generalized advancing layers method is presented here as a method of generating meshes suitable for capturing such flows. The method includes several new technical advances allowing it to mesh complex geometric domains that cannot be handled by other techniques. It is currently being used for simulations in the automotive industry. Copyright

A stabilized mixed finite element method for finite elasticity.: Formulation for linear displacement and pressure interpolation

Abstract A stabilized mixed finite element method for finite elasticity is presented. The method circumvents the fulfillment of the Ladyzenskaya–Babuska–Brezzi condition by adding mesh-dependent terms, which are functions of the residuals of the Euler–Lagrange equations, to the usual Galerkin method. The weak form and the linearized weak form are presented in terms of the reference and current configuration. Numerical experiments using a tetrahedral element with linear shape functions for the displacements and for the pressure show that the method successfully yields a stabilized element.