Simulation of elastic wave propagation in geological media: Intercomparison of three numerical methods

For wave propagation in heterogeneous media, we compare numerical results produced by grid-characteristic methods on structured rectangular and unstructured triangular meshes and by a discontinuous Galerkin method on unstructured triangular meshes as applied to the linear system of elasticity equations in the context of direct seismic exploration with an anticlinal trap model. It is shown that the resulting synthetic seismograms are in reasonable quantitative agreement. The grid-characteristic method on structured meshes requires more nodes for approximating curved boundaries, but it has a higher computation speed, which makes it preferable for the given class of problems.     

Mathematical simulation of deformation and wave processes in multilayered structures

The deformation and wave processes induced by collisions of an impactor with deformable layered targets of various configurations are analyzed. The numerical solution of such problems is associated with an adequate treatment of wave processes in a continuous medium, which is an especially difficult task in the case of layered targets. To deal with the former problem, it is proposed to use adaptive Lagrangian triangular meshes. Wave processes are simulated using the grid-characteristic method, which can serve as a basis for algorithms that do not fail near the boundary of the computational domain and at numerous material interfaces. Additionally, hybrid and hybridized grid-characteristic schemes are applied that substantially improve numerical solutions with steep gradients (discontinuous solutions). These methods provide an adequate treatment of wave processes in layered targets (wave reflection and refraction at contact surfaces, secondary-wave interaction, changes in the conditions on these boundaries, etc.).     

A study of high-order grid-characteristic methods on unstructured grids

In this paper, grid-characteristic methods for solving hyperbolic systems using approximation with high-order interpolation on unstructured tetrahedral and triangular grids are studied. Interpolation methods of order from 1 through 5 inclusive are considered. One-dimensional finite difference schemes for these methods are presented. The stability of these schemes is investigated. The grid-characteristic methods on unstructured triangular and tetrahedral grids are successfully used in seismic prospecting, specifically under the Arctic shelf and permafrost conditions. They are also used to solve problems of seismics, dynamic deformation and destruction, and to study anisotropic composite materials.     

The asymptotically optimal meshsize function for bi-p degree interpolation over rectangular elements

A method is presented to recover near optimal interpolation on finite element meshes based on information in the approximation error on an initial mesh. Only a certain class of admissable meshes with rectangular elements in the computational domains are allowed. The method attempts to reach the optimal mesh in one step from the initial mesh, and is based on the notion of meshsize function components or mesh density functions. Asymptotical results showing the optimality of the recovered meshes are given, and extensive computational verification of the method in the special case of Lagrange polynomial interpolation is provided.     

Node-based smoothed radial point interpolation method for electromagnetic-thermal coupled analysis

The node-based smoothed radial point interpolation method for solving the transient responses of magneto-electro-elastic structures in thermal environment is proposed. Considering the coupling relations between the elasticity, magnetism, electricity and heat, the generalized displacement (displacement, electric potential and magnetic potential) is calculated using the modified Newmark method. G space theory and the weakened weak formulation are applied to derive the equations of node-based smoothed radial point interpolation method for the magneto-electro-thermo-elastic multi-physics coupling problems. We use triangular background meshes as they could be generated more easily for structures with complex geometry. In some cases, they could even be created automatically. Detailed numerical study has shown that node-based smoothed radial point interpolation method not only successfully overcomes the overly-stiff behavior in the FEM and provides more accurate results, but also works well with distorted meshes. Therefore, the node-based smoothed radial point interpolation method could be adopted to solve the magneto-electro-thermo-elastic multi-physics coupling problems in practical engineering.     

An Improved Linearity-Preserving Cell-Centered Scheme for Nonlinear Diffusion Problems on General Meshes

In this paper, we suggest a new vertex interpolation algorithm to improve an existing cell-centered finite volume scheme for nonlinear diffusion problems on general meshes. The new vertex interpolation algorithm is derived by applying a special limit procedure to the well-known MPFA-O method. Since the MPFA-O method for 3D cases has been addressed in some studies, the new vertex interpolation algorithm can be extended to 3D cases naturally. More interesting is that the solvability of the corresponding local system is proved under some assumptions. Additionally, we modify the edge flux approximation by an edge-based discretization of diffusion coefficient, and thus the improved scheme is free of the so-called numerical heat-barrier issue suffered by many existing cell-centered or hybrid schemes. The final scheme allows arbitrary continuous or discontinuous diffusion coefficients and can be applicable to arbitrary star-shaped polygonal meshes. A second-order convergence rate for the approximate solution and a first-order accuracy for the flux are observed in numerical experiments. In the comparative experiments with some existing vertex interpolation algorithms, the new algorithm shows obvious improvement on highly distorted meshes.     

Piecewise parabolic method on local stencil for gasdynamic simulations

A numerical method based on piecewise parabolic difference approximations is proposed for solving hyperbolic systems of equations. The design of its numerical scheme is based on the conservation of Riemann invariants along the characteristic curves of a system of equations, which makes it possible to discard the four-point interpolation procedure used in the standard piecewise parabolic method (PPM) and to use the data from the previous time level in the reconstruction of the solution inside difference cells. As a result, discontinuous solutions can be accurately represented without adding excessive dissipation. A local stencil is also convenient for computations on adaptive meshes. The new method is compared with PPM by solving test problems for the linear advection equation and the inviscid Burgers equation. The efficiency of the methods is compared in terms of errors in various norms. A technique for solving the gas dynamics equations is described and tested for several one-and two-dimensional problems.     

Combined grid-characteristic method for the numerical solution of three-dimensional dynamical elastoplastic problems

A combined method blending the advantages of smoothed particles hydrodynamics (SPH) and the grid-characteristic method (GCM) is proposed for simulating elastoplastic bodies. Various grid methods, including the GCM, have long been used for the numerical simulation of elastoplastic media. This method applies to the simulation of wave processes in elastic media, including elastic impacts, in which case an advantage is the use of moving tetrahedral meshes. Additionally, fracture processes can be simulated by applying various fracture criteria. However, this is a technically complicated task with the accuracy of the results degrading due to the continual updating of the grid. A more suitable approach to the simulation of processes involving substantial fractures and deformations is based on SPH, which is a meshless method. However, this method also has shortcomings: it produces spurious modes, and the simulation of oscillations requires particle refinement. Thus, two families of methods are available that are optimal as applied to two different groups of problems. However, a realworld problem can frequently be a mixed one, which requires a substantial tradeoff in the numerical methods applied. Aimed at solving such problems, a combined GCM-SPH method is developed that blends the advantages of two constituting techniques and partially eliminates their shortcomings.     

Calculating the vertex unknowns of nine point scheme on quadrilateral meshes for diffusion equation

In the construction of nine point scheme,both vertex unknowns and cell-centered unknowns are introduced,and the vertex unknowns are usually eliminated by using the interpolation of neighboring cell-centered unknowns,which often leads to lose accuracy.Instead of using interpolation,here we propose a different method of calculating the vertex unknowns of nine point scheme,which are solved independently on a new generated mesh.This new mesh is a Vorono¨i mesh based on the vertexes of primary mesh and some additional points on the interface.The advantage of this method is that it is particularly suitable for solving diffusion problems with discontinuous coeffcients on highly distorted meshes,and it leads to a symmetric positive definite matrix.We prove that the method has first-order convergence on distorted meshes.Numerical experiments show that the method obtains nearly second-order accuracy on distorted meshes.     

Parallel Algorithms for Orthotropic Problems

Finite element meshes and node-numberings suitable for parallel solution with equally loaded processors are presented for linear orthotropic elliptic partial differential equations. These problems are of great importance, for instance in the oil and airfoil industries. The linear systems of equations are solved by the conjugate gradient method preconditioned by modified incomplete factorization, MIC. The basic method presented, is based on fronts of uncoupled nodes and unlike earlier methods it has the advantage of no requirement of a specific orientation of the mesh. This method is however, in general, restricted to small degree of anisotropy in the differential equation. Another method, which does not suffer from this limitation, uses rotation of the differential equation and spectral equivalence. The rotation is made in such a way that in the new co-ordinate system, the basic method is applicable. The spectral equivalence property is used for estimation of the condition number of the preconditioned system. Both methods are suitable for implementation on parallel computers. The computer architecture could be single instruction multiple data (SIMD) as well as multiple instruction multiple data (MIMD) with shared or distributed memory. Implementation of the basic method on a shared memory parallel computer shows a significant improvement by use of the MIC method compared with the diagonal scaling preconditioning method.     

Finite Difference Method with Nonuniform Meshes for Quasilinear Parabolic Systems

1.lnthestudyoftheprobleminphysics,mechanics,chemicalreactions,biologyandotherpracticalsciences,thelinearandnonlinearparabolicequationsandsystemsareappearedveryfrequently.Manynumericalinvestigationsinscientificandengineeringproblemsespeciallyinthelargescalecomputationalproblemsoftencontainthenumer-icalsolutionsofparabolicequationsandsystems.ThemethodwithunequalmeshstePSisnotavoidableinthesecomputations.Manyunexpectedandselfcontradictoryphe-nomenonraisingfromtheuseofunequalmeshstepscallourgreata…     

High-performance computer simulation of wave processes in geological media in seismic exploration

A class of problems arising in seismic exploration are investigated, namely, seismic signal propagation in multilayered geological rock and near-surface disturbance propagation in massive rock with heterogeneities, such as empty or filled fractures and cavities. Numerical solutions are obtained for wave propagation in such highly heterogeneous media, including those taking into account the plastic properties of the rock, which can be manifested near a seismic gap or a wellbore. All types of explosion-generated elastic and elastoplastic waves and waves reflected from fractures and the boundaries of the integration domain are analyzed. The identification of waves in seismograms recorded with near-surface receivers is addressed. The grid-characteristic method is used on triangular, parallelepipedal, and tetrahedral meshes with boundary conditions set on the rock-fracture interface and on free surfaces in explicit form. The numerical method proposed is suitable for the study of the interaction between seismic waves and heterogeneous inclusions, since it ensures the most correct design of computational algorithms on the boundaries of the integration domain and at media interfaces. A parallel software code implemented with the help of OpenMP and MPI was used to execute computations on parallelepipedal and tetrahedral grids.     

有限元超收敛新论

本文从三个方面讨论二阶椭圆问题有限元超收敛．1．一致网格上的新超收敛结果．利用新的“投影型插值”,我们解决了高次三角形元的超收敛问题．2．一般网格的超收敛性．利用局部插值处理和局部磨光处理我们获得了整体超收敛性结果．3．关于当前的两种超收敛技巧．Cornell学派利用一个精致的内估计和网格的点对称性,获得了一个“普遍”的结果,中国学派利用两个基本估计和离散Green函数理论获得了令人满意的结果,两者均很复杂．本文综合了两个学派的方法,简洁地证得上述普遍结果．     

Application of a fractional steps method for the numerical solution of the two-dimensional modeling of the Lake Mariut

A fractional steps technique for the numerical solution of the shallow water equations is applied to study the water velocity in Lake Mariut, its concentration and the distribution of the temperature along it. Lake Mariut is considering the most productive natural systems in Egypt. The current configuration of this lake is changing rapidly, due to people’s activities and natural processes. Most of its water supply comes from polluted agricultural drains. Several problems affect the conservation of the Lake Mariut, mainly pollution, land reclamation, intensive aquatic vegetation, over fishing and coastal erosion. The shallow water equations for this lake are discretized on a fixed grid and time stepped with the fractional steps method, where the Riemann invariants of the equations are interpolated at each time step along the characteristics of the equations using a cubic spline interpolation. The method is efficient and simple, since it evolves the equations without the iterative steps involved in the multi-dimensional interpolation problem. The absence of iterative steps in the present technique makes it very suitable for the problems in which small time steps and grid sizes are required and the simplicity of the method makes it very suitable for parallel computer. Therefore, the method provides numerical algorithms which are more efficient than other classical schemes.     

General Interpolation Formulas for Spaces of Discrete Functions with Nonuniform Meshes

1.IntroductionThegreatnumber0fproblemsf0rthelargescalescientificandengineeringcompu-tati0nsc0ncernthenumericalsolutionsofvari0uspr0blemsforthepartialdifferentialequationsandsystemsinmathematicalphysics.Thefinitedifferencemethodisthem0stc0mm0nlyusedinthesec0mputati0ns.S0thethe0reticalandnumericalstudiesofthefinitedifferenceschemesforthepr0blems0fthepartialdifferentialequati0nsandsystemsnaturallycallpeople'sgreatattentions.Theimbeddingthe0remsandtheinterpolationformulasf0rthefunctions0fS0b0lev'…     

Unconditional Superconvergent Analysis of Quasi-Wilson Element for Benjamin-Bona-Mahoney Equation

This article aims to study the unconditional superconvergent behavior of nonconforming quadrilateral quasi-Wilson element for nonlinear Benjamin Bona Mahoney (BBM) equation. For the generalized rectangular meshes including rectangular mesh, deformed rectangular mesh and piecewise deformed rectangular mesh, by use of the special character of this element, that is, the conforming part (bilinear element) has high accuracy estimates on the generalized rectangular meshes and the consistency error can reach order $O(h^2)$, one order higher than its interpolation error, the superconvergent estimates with respect to mesh size $h$ are obtained in the broken $H^1$-norm for the semi-/ fully-discrete schemes. A striking ingredient is that the restrictions between mesh size $h$ and time step $\tau$ required in the previous works are removed. Finally, some numerical results are provided to confirm the theoretical analysis.     

On bicompact grid-characteristic schemes for the linear advection equation

A set of grid-characteristic schemes for the linear advection equation is considered. Depending on the behavior of the solution, hybrid compact difference schemes of second–third order accuracy are proposed as based on interpolation polynomials. The schemes produce monotone solutions and only slightly smear discontinuities.     

Stokes问题的各向异性平行四边形有限元逼近

1 引言 Stokes问题是标准的混合问题,速度与压力同时计算,关于该问题有限元求解的文章很多(见文献[1-5])但大多都是基于对区域的正则剖分或拟一致剖分,即要求网格剖分满足hk/pK≤C,(A)K∈Jh,其中C>0为一常数,hk,pK分别为单元K的直径及内切园直径,在实际应用问题中,由于边界层或区域的拐角处需考虑物质的各向异性特征,此时对空间区域Q的剖分不再满足正则性或拟一致条件,而需要用各向异性网格剖分,才能更贴切地描述其真实情形.     

On the use of Mühlbach expansions in the recovery step of ENO methods

Summary. The recovery step is the most expensive algorithmic ingredient in modern essentially non-oscillatory (ENO) shock capturing methods on triangular meshes for the numerical simulation of compressible fluid flow. While recovery polynomials in Newton form are used in one-dimensional ENO schemes it is a priori not clear whether such useful as well as numerically stable form of polynomials exists in multiple dimensions. As was observed in [1] a very general answer to this question was provided by Mühlbach in two subsequent papers [15] and [16]. We generalise his interpolation theory further to the general recovery problem and outline the use of Mühlbach's expansion in ENO schemes. Numerical examples show the usefulness of this approach in the problem of recovery from cell average data. Received August 24, 1995 / Revised version received December 14, 1995