General Three-dimensional Domains Research Articles

An FC-based spectral solver for elastodynamic problems in general three-dimensional domains

This paper presents a spectral numerical algorithm for the solution of elastodynamics problems in general three-dimensional domains. Based on a recently introduced “Fourier continuation” (FC) methodology for accurate Fourier expansion of non-periodic functions, the proposed approach possesses a number of appealing properties: it yields results that are essentially free of dispersion errors, it entails mild CFL constraints, it runs at a cost that scales linearly with the discretization sizes, and it lends itself easily to efficient parallelization in distributed-memory computing clusters. The proposed algorithm is demonstrated in this paper by means of a number of applications to problems of isotropic elastodynamics that arise in the fields of materials science and seismology. These examples suggest that the new approach can yield solutions within a prescribed error tolerance by means of significantly smaller discretizations and shorter computing times than those required by other methods.

Asymptotic analysis of narrow escape problems in nonspherical three-dimensional domains.

Narrow escape problems consider the calculation of the mean first passage time (MFPT) for a particle undergoing Brownian motion in a domain with a boundary that is everywhere reflecting except for at finitely many small holes. Asymptotic methods for solving these problems involve finding approximations for the MFPT and average MFPT that increase in accuracy with decreasing hole sizes. While relatively much is known for the two-dimensional case, the results available for general three-dimensional domains are rather limited. This paper addresses the problem of finding the average MFPT for a class of three-dimensional domains bounded by the level surface of an orthogonal coordinate system. In particular, this class includes spheroids and other solids of revolution. The primary result presented is a two-term asymptotic expansion for the average MFPT of such domains containing an arbitrary number of holes. Steps are taken towards finding higher-order asymptotic expansions for both the average MFPT and the MFPT in these domains. The results for the average MFPT are compared to full numerical calculations performed with the comsol Multiphysics finite element solver for three distinct domains: prolate and oblate spheroids and biconcave disks. This comparison shows good agreement with the proposed two-term expansion of the average MFPT in the three domains.

On elastic solids with limiting small strain: modelling and analysis

In order to understand nonlinear responses of materials to external stimuli of different sort, be they of mechanical, thermal, electrical, magnetic, or of optical nature, it is useful to have at one's disposal a broad spectrum of models that have the capacity to describe in mathematical terms a wide range of material behavior. It is advantageous if such a framework stems from a simple and elegant general idea. Implicit constitutive theory of materials provides such a framework: while being built upon simple ideas, it is able to capture experimental observations with the minimum number of physical quantities involved. It also provides theoretical justification in the full three-dimensional setting for various models that were previously proposed in an ad hoc manner. From the perspective of the theory of nonlinear partial differential equations, implicit constitutive theory leads to new classes of challenging mathematical problems. This study focuses on implicit constitutive models for elastic solids in general, and on its subclass consisting of elastic solids with limiting small strain. After introducing the basic concepts of implicit constitutive theory, we provide an overview of results concerning modeling within the framework of continuum mechanics. We then concentrate on the mathematical analysis of relevant boundary-value problems associated with models with limiting small strain, and we present the first analytical result concerning the existence of weak solutions in general three-dimensional domains.

Computing edge stress intensity functions (ESIFs) along circular 3-D edges

A newly developed method, named the quasi-dual function method (QDFM) is proposed for extracting edge stress intensity functions (ESIFs) along circular crack fronts from finite element solutions, in a general three-dimensional domain and boundary conditions. The mathematical machinery developed in the framework of the Laplace operator in Shannon et al. (2013) is extended here to the elasticity system and applied for the extraction of ESIFs from high-order finite element solutions.The QDFM has several important advantages: (a) It allows to extract the ESIFs away from the singular edge, thus avoiding the need for a refined FE mesh, (b) The ESIFs are obtained as a function along the edge and not as pointwise values, and (c) The method is general in the sense that it is applicable to any circular edge (be it a penny shaped crack, a cylindrical crack or a circular external crack). Numerical examples are provided that demonstrate the efficiency, robustness and high accuracy of the proposed QDFM.

A quality and efficient tetrahedral mesh generation method

A method of unstructured tetrahedral mesh generation for general three-dimensional domain of arbitrary shape is presented. The method consists of two steps. First, generating the boundary tetrahedral mesh by the classic Delaunay method, then, using the advancing front method to create internal points and inserting them into the mesh by the Delaunay kernel procedure. The algorithm combines the advantages of the high point placement quality of the advancing front method and the high efficiency and convergence guaranty of the Delaunay algorithm. To get better efficiency, a new front identification and field point generation method is proposed and applied. Several examples have demonstrated the computational efficiency of our method and the high quality of meshes that generated within a reasonable time limit.

Local Solutions of a Compressible Flow Problem with Navier Boundary Conditions in General Three-Dimensional Domains

We prove the local-in-time existence of smooth solutions of the Navier–Stokes equations of compressible, barotropic flow posed in a bounded set in $\mathbb{R}^3$ with a mixed boundary condition sufficiently general to include the condition proposed by Navier that on the boundary of the spatial domain the velocity should be proportional to the projection of the stress onto the tangent plane.

Automatic p-version mesh generation for curved domains

To achieve the exponential rates of convergence possible with the p-version finite element method requires properly constructed meshes. In the case of piecewise smooth domains, these meshes are characterized by having large curved elements over smooth portions of the domain and geometrically graded curved elements to isolate the edge and vertex singularities that are of interest. This paper presents a procedure under development for the automatic generation of such meshes for general three-dimensional domains defined in solid modeling systems. Two key steps in the procedure are the determination of the singular model edges and vertices, and the creation of geometrically graded elements around those entities. The other key step is the use of general curved element mesh modification procedures to correct any invalid elements created by the curving of mesh entities on the model boundary, which is required to ensure a properly geometric approximation of the domain. Example meshes are included to demonstrate the features of the procedure.

Tetrahedral mesh generation based on node insertion in crystal lattice arrangements and advancing-front-Delaunay triangulation

A method of unstructured technical mesh generation for general three-dimensional domains is presented. A conventional boundary representation is adopted as the basis for the description of solids with evolving geometry and topology. The geometry of the surfaces is represented either analytically or by piecewise polynominal interpolation. A preliminary surface mesh is generated by an advancing-front method, with the nodes inserted by hard-sphere packing in physical space in accordance with a prescribed mesh density. Interior nodes are inserted in a face-centered-cubic (FCC) crystal lattice arrangements coupled to octree spatial subdivision, with the local lattice parameter determined by a prespecified nodal density function. Prior to triangulation of the volume, the preliminary surface mesh is preprocessed by a combination of local transformations and subdivisions in order to guarantee that the surface triangulation be a subcomplex of the volume Delaunay triangulation. A joint Delaunay triangulation of the interior and boundary nodes which preserves the modified surface mesh is then constructed via an advancing-front approach. The resulting mesh is finally improved upon by the application of local transformations. The overall time complexity of the mesher is O( N log N). The robustness and versatility of the approach, as well as the good quality of the resulting meshes, is demostrated with the aid of selected examples.

A Stable Penalty Method for the Compressible Navier--Stokes Equations: II. One-Dimensional Domain Decomposition Schemes

This paper, concluding the trilogy, develops schemes for the stable solution of wave-dominated unsteady problems in general three-dimensional domains. The schemes utilize a spectral approximation in each subdomain and asymptotic stability of the semidiscrete schemes is established. The complex computational domains are constructed by using nonoverlapping quadrilaterals in the two-dimensional case and hexahedrals in the three-dimensional space. To illustrate the ideas underlying the multidomain method, a stable scheme for the solution of the three-dimensional linear advection-diffusion equation in general curvilinear coordinates is developed. The analysis suggests a novel, yet simple, stable treatment of geometric singularities like edges and vertices. The theoretical results are supported by a two-dimensional implementation of the scheme. The main part of the paper is devoted to the development of a spectral multidomain scheme for the compressible Navier--Stokes equations on conservation form and a unified approach for dealing with the open boundaries and subdomain boundaries is presented. Well posedness and asymptotic stability of the semidiscrete scheme is established in a general curvilinear volume, with special attention given to a hexahedral domain. The treatment includes a stable procedure for dealing with boundary conditions at a solid wall. The efficacy of the scheme for the compressible Navier--Stokes equations is illustrated by obtaining solutions to subsonic and supersonic boundary layer flows with various types of boundary conditions. The results are found to agree with the solution of the compressible boundary layer equations.

Update to: Approaches to the Automatic Generation and Control of Finite Element Meshes

This paper updates the status of efforts on the development of automatic mesh generation techniques for general three-dimensional domains. The technical areas reviewed include: (i) issues associated with automatic mesh generation of CAD geometric models, (ii) local mesh modification procedures for improving mesh quality, (iii) advances in tetrahedral mesh generators, (iv) generation of anisotropic meshes, (v) hexahedral mesh generators, and (vi) implementation of automatic mesh generators on parallel computers.