Sheng-Tao John Yu

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.

Numerical Solution by the CESE Method of a First-Order Hyperbolic Form of the Equations of Dynamic Nonlinear Elasticity

This paper reports the application of the space-time conservation element and solution element (CESE) method to the numerical solution of nonlinear waves in elastic solids. The governing equations consist of a pair of coupled first-order nonlinear hyperbolic partial differential equations, formulated in the Eulerian frame. We report their derivations and present conservative, nonconservative, and diagonal forms. The conservative form is solved numerically by the CESE method; the other forms are used to study the eigenstructure of the hyperbolic system (which reveals the underlying wave physics) and deduce the Riemann invariants. The proposed theoretical/numerical approach is demonstrated by directly solving two benchmark elastic wave problems: one involving linear propagating extensional waves, the other involving nonlinear resonant standing waves. For the extensional wave problem, the CESE method accurately captures the sharp propagating wavefront without excessive numerical diffusion or spurious oscillations, and predicts correct reflection characteristics at the boundaries. For the resonant vibrations problem, the CESE method captures the linear-to-nonlinear evolution of the resonant waves and the distribution of wave energy among multiple modes in the nonlinear regime.

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.

A 3D Unstructured Mesh Euler Solver Based on the Fourth-Order CESE Method

Abstract : In this paper, the CESE method is extended and employed to construct a fourth-order, three-dimensional, unstructured-mesh solver for hyperbolic Partial Differential Equations (PDEs). This new CESE method retains all favorable attributes of the original second-order CESE method, including: (i) flux conservation in space and time without using a one-dimensional Riemann solver, (ii) genuinely multi-dimensional treatment without dimensional splitting (iii) the CFL constraint remains to be less than or equal to 1, and (iv) the use of a compact mesh stencil involving only the immediate neighboring nodes surrounding the node where the solution is sought. Two validation cases are presented. First higher order convergence is demonstrated by the linear advection equation. Second supersonic flow over a spherical body is simulated to demonstrates the schemes ability to accurately resolve discontinuities.

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.