A stable discontinuous Galerkin method for the perfectly matched layer for elastodynamics in first order form

We present a stable discontinuous Galerkin (DG) method with a perfectly matched layer (PML) for three and two space dimensional linear elastodynamics, in velocity-stress formulation, subject to well-posed linear boundary conditions. First, we consider the elastodynamics equation, in a cuboidal domain, and derive an unsplit PML truncating the domain using complex coordinate stretching. Leveraging the hyperbolic structure of the underlying system, we construct continuous energy estimates, in the time domain for the elastic wave equation, and in the Laplace space for a sequence of PML model problems, with variations in one, two and three space dimensions, respectively. They correspond to PMLs normal to boundary faces, along edges and in corners. Second, we develop a DG numerical method for the linear elastodynamics equation using physically motivated numerical flux and penalty parameters, which are compatible with all well-posed, internal and external, boundary conditions. When the PML damping vanishes, by construction, our choice of penalty parameters yield an upwind scheme and a discrete energy estimate analogous to the continuous energy estimate. Third, to ensure numerical stability of the discretization when PML damping is present, it is necessary to extend the numerical DG fluxes, and the numerical inter-element and boundary procedures, to the PML auxiliary differential equations. This is crucial for deriving discrete energy estimates analogous to the continuous energy estimates. By combining the DG spatial approximation with the high order ADER time stepping scheme and the accuracy of the PML we obtain an arbitrarily accurate wave propagation solver in the time domain. Numerical experiments are presented in two and three space dimensions corroborating the theoretical results

ExaHyPE: An engine for parallel dynamically adaptive simulations of wave problems

ExaHyPE (“An Exascale Hyperbolic PDE Engine”) is a software engine for solving systems of first-order hyperbolic partial differential equations (PDEs). Hyperbolic PDEs are typically derived from the conservation laws of physics and are useful in a wide range of application areas. Applications powered by ExaHyPE can be run on a student’s laptop, but are also able to exploit thousands of processor cores on state-of-the-art supercomputers. The engine is able to dynamically increase the accuracy of the simulation using adaptive mesh refinement where required. Due to the robustness and shock capturing abilities of ExaHyPE’s numerical methods, users of the engine can simulate linear and non-linear hyperbolic PDEs with very high accuracy. Users can tailor the engine to their particular PDE by specifying evolved quantities, fluxes, and source terms. A complete simulation code for a new hyperbolic PDE can often be realised within a few hours — a task that, traditionally, can take weeks, months, often years for researchers starting from scratch. In this paper, we showcase ExaHyPE’s workflow and capabilities through real-world scenarios from our two main application areas: seismology and astrophysics

The EU Center of Excellence for Exascale in Solid Earth (ChEESE): Implementation, results, and roadmap for the second phase

publishedVersio

Vectorization and Minimization of Memory Footprint for Linear High-Order Discontinuous Galerkin Schemes

We present a sequence of optimizations to the performance-critical compute kernels of the high-order discontinuous Galerkin solver of the hyperbolic PDE engine ExaHyPE -- successively tackling bottlenecks due to SIMD operations, cache hierarchies and restrictions in the software design. Starting from a generic scalar implementation of the numerical scheme, our first optimized variant applies state-of-the-art optimization techniques by vectorizing loops, improving the data layout and using Loop-over-GEMM to perform tensor contractions via highly optimized matrix multiplication functions provided by the LIBXSMM library. We show that memory stalls due to a memory footprint exceeding our L2 cache size hindered the vectorization gains. We therefore introduce a new kernel that applies a sum factorization approach to reduce the kernel's memory footprint and improve its cache locality. With the L2 cache bottleneck removed, we were able to exploit additional vectorization opportunities, by introducing a hybrid Array-of-Structure-of-Array data layout that solves the data layout conflict between matrix multiplications kernels and the point-wise functions to implement PDE-specific terms. With this last kernel, evaluated in a benchmark simulation at high polynomial order, only 2\% of the floating point operations are still performed using scalar instructions and 22.5\% of the available performance is achieved.Comment: PDSEC 202

Earthquake Rupture on Multiple Splay Faults and Its Effect on Tsunamis

Detailed imaging of accretionary wedges reveals splay fault networks that could pose a significant tsunami hazard. However, the dynamics of multiple splay fault activation during megathrust earthquakes and the consequent effects on tsunami generation are not well understood. We use a 2-D dynamic rupture model with complex topo-bathymetry and six curved splay fault geometries constrained from realistic tectonic loading modeled by a geodynamic seismic cycle model with consistent initial stress and strength conditions. We find that all splay faults rupture coseismically. While the largest splay fault slips due to a complex rupture branching process from the megathrust, all other splay faults are activated either top down or bottom up by dynamic stress transfer induced by trapped seismic waves. We ascribe these differences to local non-optimal fault orientations and variable along-dip strength excess. Generally, rupture on splay faults is facilitated by their favorable stress orientations and low strength excess as a result of high pore-fluid pressures. The ensuing tsunami modeled with non-linear 1-D shallow water equations consists of one high-amplitude crest related to rupture on the longest splay fault and a second broader wave packet resulting from slip on the other faults. This results in two episodes of flooding and a larger run-up distance than the single long-wavelength (300 km) tsunami sourced by the megathrust-only rupture. Since splay fault activation is determined by both variable stress and strength conditions and dynamic activation, considering both tectonic and earthquake processes is relevant for understanding tsunamigenesis.ISSN:2169-9313ISSN:0148-0227ISSN:2169-935

A stable discontinuous Galerkin method for linear elastodynamics in 3D geometrically complex elastic solids using physics based numerical fluxes

Time-stable, high order accurate and explicit numerical methods are effective for hyperbolic wave propagation problems. As a result of the complexities of real geometries, internal interfaces and nonlinear boundary and interface conditions, discontinuities and sharp wave fronts may become fundamental features of the solution. Therefore, geometrically flexible and adaptive numerical algorithms are crucial for high fidelity and efficient simulations of wave phenomena in many applications. Adaptive curvilinear meshes hold promise to minimise the effort to represent complicated geometries or heterogeneous material data avoiding the bottleneck of feature-preserving meshing. To enable the design of stable DG methods on three space dimensional (3D) curvilinear elements we construct a structure preserving skew-symmetric coordinate transformation motivated by the underlying physics. Using a physics-based numerical penalty-flux, we develop a 3D provably energy-stable discontinuous Galerkin finite element approximation of the elastic wave equation in geometrically complex and heterogeneous media. By construction, our numerical flux is upwind and yields a discrete energy estimate analogous to the continuous energy estimate. The ability to treat conforming and non-conforming curvilinear elements allows for flexible adaptive mesh refinement strategies. The numerical scheme has been implemented in ExaHyPE, a simulation engine for parallel dynamically adaptive simulations of wave problems on adaptive Cartesian meshes. We present 3D numerical experiments of wave propagation in heterogeneous isotropic and anisotropic elastic solids demonstrating stability and high order accuracy. We demonstrate the potential of our approach for computational seismology in a regional wave propagation scenario in a geologically constrained 3D model including the geometrically complex free-surface topography of Mount Zugspitze, Germany.ISSN:0045-7825ISSN:1879-213

High performance uncertainty quantification with parallelized multilevel Markov chain Monte Carlo

Numerical models of complex real-world phenomena often necessitate High Performance Computing (HPC). Uncertainties increase problem dimensionality further and pose even greater challenges. We present a parallelization strategy for multilevel Markov chain Monte Carlo, a state-of-the-art, algorithmically scalable Uncertainty Quantification (UQ) algorithm for Bayesian inverse problems, and a new software framework allowing for large-scale parallelism across forward model evaluations and the UQ algorithms themselves. The main scalability challenge presents itself in the form of strong data dependencies introduced by the MLMCMC method, prohibiting trivial parallelization. Our software is released as part of the modular and open-source MIT Uncertainty Quantification Library (MUQ), and can easily be coupled with arbitrary user codes. We demonstrate it using the Distributed and Unified Numerics Environment (DUNE) and the ExaHyPE Engine. The latter provides a realistic, large-scale tsunami model in which we identify the source of a tsunami from buoy-elevation data

A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography

In most classical approaches of computational geophysics for seismic wave propagation problems, complex surface topography is either accounted for by boundary-fitted unstructured meshes, or, where possible, by mapping the complex computational domain from physical space to a topologically simple domain in a reference coordinate system. However, all these conventional approaches face problems if the geometry of the problem becomes sufficiently complex. They either need a mesh generator to create unstructured boundary-fitted grids, which can become quite difficult and may require a lot of manual user interactions in order to obtain a high quality mesh, or they need the explicit computation of an appropriate mapping function from physical to reference coordinates. For sufficiently complex geometries such mappings may either not exist or their Jacobian could become close to singular. Furthermore, in both conventional approaches low quality grids will always lead to very small time steps due to the Courant-Friedrichs-Lewy (CFL) condition for explicit schemes. In this paper, we propose a completely different strategy that follows the ideas of the successful family of high resolution shock-capturing schemes, where discontinuities can actually be resolved anywhere on the grid, without having to fit them exactly. We address the problem of geometrically complex free surface boundary conditions for seismic wave propagation problems with a novel diffuse interface method (DIM) on adaptive Cartesian meshes (AMR) that consists in the introduction of a characteristic function 0≤α≤1 which identifies the location of the solid medium and the surrounding air (or vacuum) and thus implicitly defines the location of the free surface boundary. Physically, α represents the volume fraction of the solid medium present in a control volume. Our new approach completely avoids the problem of mesh generation, since all that is needed for the definition of the complex surface topography is to set a scalar color function to unity inside the regions covered by the solid and to zero outside. The governing equations are derived from ideas typically used in the mathematical description of compressible multiphase flows. An analysis of the eigenvalues of the PDE system shows that the complexity of the geometry has no influence on the admissible time step size due to the CFL condition. The model reduces to the classical linear elasticity equations inside the solid medium where the gradients of α are zero, while in the diffuse interface zone at the free surface boundary the governing PDE system becomes nonlinear. We can prove that the solution of the Riemann problem with arbitrary data and a jump in α from unity to zero yields a Godunov-state at the interface that satisfies the free-surface boundary condition exactly, i.e. the normal stress components vanish. In the general case of an interface that is not aligned with the grid and which is not infinitely thin, a systematic study on the distribution of the volume fraction function inside the interface and the sensitivity with respect to the thickness of the diffuse interface layer has been carried out. In order to reduce numerical dissipation, we use high order discontinuous Galerkin (DG) finite element schemes on adaptive AMR grids together with a second order accurate high resolution shock capturing subcell finite volume (FV) limiter in the diffuse interface region. We furthermore employ a little dissipative HLLEM Riemann solver, which is able to resolve the steady contact discontinuity associated with the volume fraction function and the spatially variable material parameters exactly. While the method is locally high order accurate in the regions without limiter, the global order of accuracy of the scheme is at most two if the limiter is activated. It is locally of order one inside the diffuse interface region, which is typical for shock-capturing schemes at shocks and contact discontinuities. We show a large set of computational results in two and three space dimensions involving complex geometries where the physical interface is not aligned with the grid or where it is is even moving. For all test cases we provide a quantitative comparison with classical approaches based on boundary-fitted unstructured meshes