We introduce an efficient and accurate staggered-grid finite-difference (SGFD) method to solve the two-dimensional elastic wave equation. We use a coupled first-order stress-velocity formulation. In the standard implementation of SGFD method the same SGFD operator is used to approximate the spatial derivatives. However, we propose a numerical method based on mixed SGFD operators which happen to be more efficient with similar accuracy in comparison to uniform SGFD operator. We refer the proposed method as the non-balanced SGFD numerical scheme which means combining high-order SGFD operators with second-order SGFD operators. A very care attention is directed at the derivation of the SGFD operator coefficients. The correctness of proposed scheme is proven by dispersion analysis. Through SGFD modeling examples, we verify/demonstrate that the proposed non-balanced operator offers a similar level of accuracy with a cheaper computation cost compared to the more expensive balanced SGFD method.
We present an approach to simulate the 3D isotropic elastic wave propagation using nonuniform finite difference discretization on staggered grids. Specifically, we consider simulation domains composed of layers of uniform grids with different grid spacings, separated by nonconforming interfaces. We demonstrate that this layer-wise finite difference discretization has the potential to significantly reduce the simulation cost, compared to its fully uniform counterpart. Stability of such a discretization is achieved by using specially designed difference operators, which are variants of the stand…
A Nonlinear Finite Element Heterogeneous Multiscale Method for the Homogenization of Hyperelastic Solids and a Novel Staggered Two-Scale Solution Algorithm
An iterative staggered scheme for phase field brittle fracture propagation with stabilizing parameters
University of Bergen（卑尔根大学）；Leibniz University Hannover, Institute of Applied Mathematics（汉诺威莱布尼茨大学应用数学研究所）；NORCE Norwegian Research Centre AS（挪威研究中心有限公司）；Princeton University（普林斯顿大学）
When describing the deflagration-to-detonation transition in solid granular explosives mixed with gaseous products of combustion, a well-developed two-phase mixture model is the compressible Baer-Nunziato (BN) model, containing solid and gas phases. If this model is numerically simulated by a conservative Godunov-type scheme, spurious oscillations are likely to generate from porosity interfaces, which may result from the average process of conservative variables that violates the continuity of Riemann invariants across porosity interfaces. In order to suppress the oscillations, this paper prop…
In this paper we propose and analyze a staggered discontinuous Galerkin method for a five-field formulation of the Biot system of poroelasticity on general polygonal meshes. Elasticity is equipped with stress-displacement-rotation formulation with weak stress symmetry for arbitrary polynomial orders, which extends the piecewise constant approximation developed in (L. Zhao and E.-J. Park, SIAM J. Sci. Comput. 42 (2020), A2158-A2181). The proposed method is locking free and can handle highly distorted grids possibly including hanging nodes, which is desirable for practical applications. We prove…
We present a new numerical approach for wave induced dynamic fracture. The method is based on a discontinuous Galerkin approximation of the first-order hyperbolic system for elastic waves and a phase-field approximation of brittle fracture driven by the maximum tension. The algorithm is staggered in time and combines an implicit midpoint rule for the wave propagation followed by an implicit Euler step for the phase-field evolution. At fracture, the material is degraded, and the waves are reflected at the diffusive interfaces. Two and three-dimensional examples demonstrate the advantages of the…
We present a novel staggered semi-implicit hybrid FV/FE method for the numerical solution of the shallow water equations at all Froude numbers on unstructured meshes. A semi-discretization in time of the conservative Saint-Venant equations with bottom friction terms leads to its decomposition into a first order hyperbolic subsystem containing the nonlinear convective term and a second order wave equation for the pressure. For the spatial discretization of the free surface elevation an unstructured mesh of triangular simplex elements is considered, whereas a dual grid of the edge-type is employ…
The correspondence between Voigt and Reuss bounds and the decoupling constraint in a two-grid staggered solution algorithm to coupled flow and deformation in heterogeneous poroelastic media
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Shell（壳牌）
We present a mechanism to explicitly couple the finite-difference discretizations of 2D acoustic and isotropic elastic wave systems that are separated by straight interfaces. Such coupled simulations allow the application of the elastic model to geological regions that are of special interest for seismic exploration studies (e.g., the areas surrounding salt bodies), while with the computationally more tractable acoustic model still being applied in the background regions. Specifically, the acoustic wave system is expressed in terms of velocity and pressure while the elastic wave system is expr…
We propose a new pressure-based structure-preserving (SP) and quasi asymptotic preserving (AP) staggered semi-implicit finite volume scheme for the unified first order hyperbolic formulation of continuum mechanics. The unified model is based on the theory of symmetric-hyperbolic and thermodynamically compatible (SHTC) systems and includes the description of elastic and elasto-plastic solids in the nonlinear large-strain regime as well as viscous and inviscid heat-conducting fluids, which correspond to the stiff relaxation limit of the model. In the absence of relaxation source terms, the homog…
A structure-preserving staggered semi-implicit four-split finite volume scheme for continuum mechanics on unstructured meshes
University of Trento（特伦托大学）；Southern University of Science and Technology（南方科技大学）；Université Savoie Mont Blanc（萨瓦-蒙布朗大学）；University of Ferrara（费拉拉大学）；University of Verona（维罗纳大学）；Inria（法国国家数字研究院）；Université de Strasbourg（斯特拉斯堡大学）
A posteriori modeling error estimates in the optimization of two-scale elastic composite materials
Institute for Applied Mathematics, Rheinische Friedrich-Wilhelms-Universität Bonn（波恩莱茵弗里德里希·威廉大学应用数学研究所）；Institute for Numerical Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn（波恩莱茵弗里德里希·威廉大学数值模拟研究所）
A Variational Bayesian Inference Theory of Elasticity and Its Mixed Probabilistic Finite Element Method for Inverse Deformation Solutions in Any Dimension
Propagation characteristics of a wave are defined by the dispersion relationship, from which the governing partial differential equation (PDE) can be recovered. PDEs are commonly solved numerically using the finite-difference (FD) method, with stencils constructed from truncated Taylor series expansions which, whilst typically providing good approximation of the PDE in the space-time domain, often differ considerably from the original partial differential in the wavenumber-frequency domain where the dispersion relationship is defined. Consequentially, stable, high-order FD schemes may not nece…
We propose a novel numerical method for the solution of the shallow water equations in different regimes of the Froude number making use of general polygonal meshes. The fluxes of the governing equations are split such that advection and acoustic-gravity sub-systems are derived, hence separating slow and fast phenomena. This splitting allows the nonlinear convective fluxes to be discretized explicitly in time, while retaining an implicit time marching for the acoustic-gravity terms. Consequently, the novel schemes are particularly well suited in the low Froude limit of the model, since no nume…
We analyse three time integration schemes for unfitted methods in fluid structure interaction. In Alghorithm 1 we propose a fully discrete monolithic algorithm with P1 P1 stabilized finite elements for the fluid problem; for this alghorithm we prove well-posedness, unconditional stability and convergence in the case of linearized problem (see Propositions 2.4.2, 2.4.3 and Theorem 3.3.7, respectively). The analysis optimal convergence rates as expected from the Euler scheme, and the supposed regularity of the solution to the continuous problem. Moreover we introduce two algorithms that allow fo…
We discuss the applicability of a unified hyperbolic model for continuum fluid and solid mechanics to modeling non-Newtonian flows and in particular to modeling the stress-driven solid-fluid transformations in flows of viscoplastic fluids, also called yield-stress fluids. In contrast to the conventional approaches relying on the non-linear viscosity concept of the Navier-Stokes theory and representation of the solid state as an infinitely rigid non-deformable solid, the solid state in our theory is deformable and the fluid state is considered rather as a "melted" solid via a certain procedure …
