Yung-Yu Chen

Velocity-Stress Equations for Waves in Solids of Hexagonal Symmetry Solved by the Space-Time CESE Method

This paper reports an extension of the space-time conservation element and solution element (CESE) method to simulate stress waves in elastic solids of hexagonal symmetry. The governing equations include the equation of motion and the constitutive equation of elasticity. With velocity and stress components as the unknowns, the governing equations are a set of 9, first-order, hyperbolic partial differential equations. To assess numerical accuracy of the results, the characteristic form of the equations is derived. Moreover, without using the assumed plane wave solution, the one-dimensional equations are shown to be equivalent to the Christoffel equations. The CESE method is employed to solve an integral form of the governing equations. Space-time flux conservation over conservation elements (CEs) is imposed. The integration is aided by the prescribed discretization of the unknowns in each solution element (SE), which in general does not coincide with a CE. To demonstrate this approach, numerical results in the present paper include one-dimensional expansion waves in a suddenly stopped rod, two-dimensional wave expansion from a point in a plane, and waves interacting with interfaces separating hexagonal solids with different orientations. All results show salient features of wave propagation in hexagonal solids and the results compared well with the available analytical solutions.

SOLVCON: A Python-Based CFD Software Framework for Hybrid Parallelization

SOLVCON is a new, open-sourced software framework for high-fidelity solutions of linear and non-linear hyperbolic partial differential equations. SOLVCON emphasizes scalability, portability, and maintainability for supercomputing by using emerging multi-core architectures. The code development effort follows Extreme Programming practices, including version control, documentation, issue tracking, user support, and frequent code releases. In SOLVCON, the Python framework includes all supportive functionalities for the work flow. For pre-processing operations, the Python framework provides parallelized mesh data input and automatically sets up domain decomposition. In calculations, the Python framework provides light-weight memory management through extensive use of pointers. Computation-intensive operations are implemented by using C and FORTRAN for high performance. The default numerical algorithm employed is the space-time Conservation Element and Solution Element (CESE) method. The code uses general unstructured meshes with mixed elements, including tetrahedra, hexahedra, prisms, and pyramids for threedimensional calculations. Hybrid parallelism includes shared- and distributed-memory parallelization. The temporal loop and the spatial loop in modern finite-volume methods are implemented in a two-layered structure in SOLVCON. Distributed-memory parallelization by domain decomposition and MPI is performed in the temporal loop. Shared-memory parallel computing by using accelerator technologies, e.g., General-Purpose Graphic Processor Unit (GPGPU), is performed in the spatial loop. More than 99% of the execution time of SOLVCON is used for number-crunching in the solver as a part of the space loop. Written in C or FORTRAN, a typical solver contains only 10% of the code statements in SOLVCON. To demonstrate the capabilities of newly developed SOLVCON, we performed CFD calculations by using 23 million elements. The code was run on a 512-core cluster. SOLVCON delivers calculations of flow variables in 11.29 million elements per second. The parallel efficiency is 70%. In the open-sourced SOLVCON, two solvers are available: (i) the Euler equations solver for compressible flows, and (ii) the velocity-stress equations solver for waves in anisotropic elastic solids. SOLVCON can be easily extended for other applications, including viscous flows, aero-acoustics, nonlinear solid mechanics, and electromagnetism. The Python framework allows fast adaption to new heterogeneous, multi-core hardware as well as further development of the code for peta-scale supercomputing.

Eigenstructure of First-Order Velocity-Stress Equations for Waves in Elastic Solids of Trigonal 32 Symmetry

This paper reports the eigenstructure of a set of first-order hyperbolic partial differential equations for modeling waves in solids with a trigonal 32 symmetry. The governing equations include the equation of motion and partial differentiation of the elastic constitutive relation with respect to time. The result is a set of nine, first-order, fully coupled, hyperbolic partial differential equations with velocity and stress components as the unknowns. Shown in the vector form, the model equations have three 9×9 coefficient matrices. The wave physics are fully described by the eigenvalues and eigenvectors of these matrices; i.e., the nontrivial eigenvalues are the wave speeds, and a part of the corresponding left eigenvectors represents wave polarization. For a wave moving in a certain direction, three wave speeds can be identified by calculating the eigenvalues of the coefficient matrix in a rotated coordinate system. In this process, without using the plane-wave solution, we recover the Christoffel matrix and thus validate the formulation. To demonstrate this approach, two- and three-dimensional slowness profiles of quartz are calculated. Wave polarization vectors for wave propagation in several compression directions as well as noncompression directions are discussed.

Velocity-Stress Equations for Wave Propagation in Anisotropic Elastic Media

where ρ is the density of the medium, w the displacement, and c[4] the fourth-order stiffness tensor [2]. Equation (1.1) has been derived based on the equation of motion in conjunction with the elastic constitutive equation. Equation (1.1) has been solved by the finite-difference methods, e.g., [21], and the time-domain finite-element methods, e.g., [33], for propagating waves, and the frequency-domain finite-element methods, e.g., [6], for normal mode analysis of standing waves.

A Two-Dimensional Fourth-Order CESE Method for the Euler Equations on Triangular Unstructured Meshes

Previously, Chang 1 reported a new high-order Conservation Element Solution Element (CESE) method for solving nonlinear, scalar, hyperbolic partial differential equations in one dimensional space. Bilyeu et al. 2 have extended Chang’s scheme for solving a onedimensional, coupled equations with an arbitrary order of accuracy. In the present paper, the one-dimensional, high-order CESE method is extended for two-dimensional unstructured meshes. A formulation is presented for solving the coupled equations with the fourthorder in accuracy. The new high-order CESE methods share many favorable attributes of the original second-order CESE method, including: (i) the use of compact mesh stencil involving only the immediate mesh nodes surrounding the node where the solution is sought, (ii) the CFL stability constraint remains to be � 1, and (iii) superb shock capturing capability without using an approximate Riemann solver. To demonstrate the formulation, four test cases are reported, including (i) solution of a two-dimensional, scalar convection equation, (ii) the solution of the linear acoustic equations, (iii) the solution of the Euler equations for waves of small amplitudes, and (iv) the solution of the Euler equations for expanding shock waves. In all calculations, unstructured triangular meshes are used. In the first three cases, convergence tests show the fourth-order accuracy of the solutions. In the last case, numerical results of the fourth-order scheme are superior than that obtained by the second-order CESE method. In this paper, we extend the one-dimensional, high-order CESE method for solving two-dimensional coupled, nonlinear hyperbolic Partial Differential Equations (PDEs) by using unstructured meshes. The new formulation is currently fourth-order in accuracy, but it can be easily extended to a higher-order scheme by including more terms in the Taylor series expansion. The tenet of the CESE method is treating space and time in a unified manner in integrating the model equations for flux conservation. In the CESE method, the space-time domain is divided into non-overlapping Conservation Elements (CEs), over which the spacetime integration is performed to enforce flux conservation. The integration is facilitated by using Solution Elements (SEs), which in general do not coincide with the CEs. The unknowns are discretized inside each SE by using a Taylor series expansion. Aided by the prescribed discretization schemes for the unknowns inside each SE, space-time flux over each surface of a CE can be calculated and the overall flux conservation over each CE can be enforced. The result is an explicit formulation for updating the unknowns for time-marching calculation. Special features of the CESE method include: (i) The unknowns are discretized by using the Taylor series expansion in both space and time. The order of the Taylor series is also the order of accuracy of the method. (ii) The method has the most compact mesh stencil, which involves only the immediate neighboring mesh points of the cell where the solution is sought. (iii) The method is an explicit scheme in time-marching calculation. The stability criterion is CFL � 1. (iv) No approximate Riemann solver is used and the scheme is simple and efficient. (v) The CESE method includes a suite of numerical algorithms,