Stability of explicit-implicit mesh partitions in time integration

The stability of time integration for a semidiscretization of structures and continua where the mesh is partitioned into subdomains integrated simultaneously by explicit and implicit methods is considered. Stability in energy is proven for linear systems subject to the Courant condition in the explicit subdomain for a central difference explicit and trapezoidal implicit integrator. Numerical examples are presented to demonstrate the procedure.     

On the role of the strong ellipticity condition in nonlinear elasticity

In this paper we show that if an elastic material is strongly elliptic on certain subdomains of the domain of the strain energy function then, provided that a mild boundedness condition (which may be regarded as restricting the class of considered materials) is satisfied, the nominal work of the deformation and the strain energy must satisfy certain growth conditions on these subdomains. Then we use the growth conditions to find upper and lower bounds for solutions to the boundary value problem of place (with dead loading) which belong to the considered subdomains. If the ellipticity condition is not satisfied at all points of the considered subdomains we show that the strain energy may satisfy certain inequalities which were shown to imply the Liapunov instability of equilibrium solutions to the boundary value problem of place and the fact that solutions to the dynamical problem cannot exist globally in time for arbitrary initial data.     

基于界面单元的子结构协调技术

该文提出一种用于协调子结构的界面单元方法。基于广义变分原理,将两子域的刚度与界面单元刚度组装成耦合结构的整体刚度矩阵,求解新的平衡方程即可得到各个耦合子域的位移。界面单元的意义是对子域引入边界力,并建立边界上平衡关系和位移协调关系。该文利用悬臂梁单轴受拉案例验证了界面单元方法的精确性。为了使得界面单元能够应用到子结构混合试验中,引入静力凝聚与BFGS方法,这样只需通过提取边界上的力与位移即可实现多子域不共节点的边界协调问题。该文最终以悬臂梁案例验证了界面单元在解决非线性静、动力加载工况下的正确性。     

Iterative coupling of FEM and BEM in 3D transient elastodynamics

A domain decomposition approach is presented for the transient analysis of three-dimensional wave propagation problems. The subdomains are modelled using the FEM and/or the BEM, and the coupling of the subdomains is performed in an iterative manner, employing a sequential Neumann–Dirichlet interface relaxation algorithm which also allows for an independent choice of the time step length in each subdomain. The approach has been implemented for general 3D problems. In order to investigate the convergence behaviour of the proposed algorithm, using different combinations of FEM and BEM subdomains, a parametric study is performed with respect to the choice of the relaxation parameters. The validity of the proposed method is shown by means of two numerical examples, indicating the excellent accuracy and applicability of the new formulation.     

Iterative coupling of FEM and BEM in 3D transient elastodynamics

A domain decomposition approach is presented for the transient analysis of three-dimensional wave propagation problems. The subdomains are modelled using the FEM and/or the BEM, and the coupling of the subdomains is performed in an iterative manner, employing a sequential Neumann–Dirichlet interface relaxation algorithm which also allows for an independent choice of the time step length in each subdomain. The approach has been implemented for general 3D problems. In order to investigate the convergence behaviour of the proposed algorithm, using different combinations of FEM and BEM subdomains, a parametric study is performed with respect to the choice of the relaxation parameters. The validity of the proposed method is shown by means of two numerical examples, indicating the excellent accuracy and applicability of the new formulation.     

Non-diffusive N + 2 degree Petrov-Galerkin methods for two-dimensional transient transport computations

A new non-diffusive Petrov-Galerkin type finite element method which uses test functions two polynomial degrees higher than the trial functions is developed for the transient convection dominated transport equation in two dimensions. The scheme uses bilinear quadrilateral finite elements for the spatial discretization and Crank-Nicolson finite differencing for the time integration. The standard product extension of very successful one-dimensional N + 2 degree upwinding functions to two dimensions is ineffective for general 2-D flow problems, especially at higher Courant numbers where cross-derivative truncation terms become important. Therefore effective N + 2 degree test functions are developed through an analysis by which the truncation error terms in the discrete nodal equation are eliminated up to fifth order. The new scheme is very effective for general 2-D flows over a wide Courant number range and eliminates the troublesome cross-derivative truncation terms. The scheme is simple and robust in that the upwinding coefficients are readily defined and only dependent on Courant number. Numerical examples illustrate the excellent behaviour of the new scheme.     

Coupling subdomains with heterogeneous time integrators and incompatible time steps

The work presented in this publication can be categorized among domain decomposition methods of the dual Schur type applied to structural dynamics. This approach leads to lower CPU times and better control of the accuracy of the time discretization and allows to take into account multi-time-scale effects which arise in transient structural dynamics. In order to consider incompatible time scales, one has to enforce continuity at the interfaces between the subdomains. Here, we propose a general formalism which enables the coupling of subdomains with their own numerical time integration scheme. The proposed method enables one to take into account possible nonlinearities which may present different time scale between the subdomains in a general manner for a wide range of time numerical scheme. This method also offers an important improvement for industrial software with easy implementation. Linear and nonlinear numerical examples are proposed in order to show the efficiency and the robustness of the method.     

An isogeometric‐meshfree coupling approach for analysis of cracks

This paper presents a new concurrent simulation approach to couple isogeometric analysis (IGA) with the meshfree method for studying of crack problems. In the present method, the overall physical domain is divided into 2 subdomains that are formulated with the IGA and meshfree method, respectively. In the meshfree subdomain, the moving least squares shape function is adopted for the discretization of the area around crack tips, and the IGA subdomain is adopted in the remaining area. Meanwhile, the interface region between the 2 subdomains is represented by coupled shape functions. The resulting shape function, which comprises both IGA and meshfree shape functions, satisfies the consistency condition, thus ensuring convergence of the method. Moreover, the meshfree shape functions augmented with the enriched basis functions to predict the singular stress fields near a crack tip are presented. The proposed approach is also applied to simulate the crack propagation under a mixed‐mode condition. Several numerical examples are studied to demonstrate the use and robustness of the proposed method.     

Flow through variably saturated soils

The equation of flow through variably saturated porous media is discretized via the Galerkin finite element formulation. The discretization is coupled with an approach for mesh generation and optimization of the node numbering scheme. Sensitivity analysis showed that the solution behavior is controlled by dimensionless quantities equivalent to Peclet and Courant numbers. For the form of equation investigated, no universal limiting values of Pe and Cr can be established because the values of these parameters depend on both the constitutive relations used and on initial conditions. For more efficient solution of the problem, a deformation scheme of the computational mesh is proposed, which accounts for the limiting Peclet and Courant numbers and for the shape of the deformed elements. Comparisons with other solutions showed that the numerical scheme performs very well.     

A BEM based domain decomposition method for nonlinear analysis of elastic space membranes

In this paper the Domain Decomposition Method (DDM) is developed for nonlinear analysis of both flat and space elastic membranes of complicated geometry which may have holes. The domain of the projection of the membrane on the xy plane is decomposed into non-overlapping subdomains and the membrane problem is solved sequentially in each subdomain starting from zero displacements on the virtual boundaries. The procedure is repeated until the traction continuity conditions are also satisfied on the virtual boundaries. The membrane problem in each subdomain is solved using the Analog Equation Method (AEM). According to this method the three coupled strongly nonlinear partial differential equations, governing the response of the membrane, are replaced by three uncoupled linear membrane equations (Poisson's equations) subjected to fictitious sources under the same boundary conditions. The fictitious sources are established using a meshless BEM procedure. Example problems are presented, for both flat and space membranes, which illustrate the method and demonstrate its efficiency and accuracy.     

Numerical solution of dispersion equation in one dimension

Abstract

An improved hybrid method for one‐dimensional advection‐diffusion problems, based on the Holly‐Preissmann two‐point fourth‐order and Crank‐Nicholson numerical schemes, has been proposed to handle the problem with Courant numbers (Cr) greater than 1. Extensive test runs and analyses have been performed for a schematic advection‐diffusion problem. Through a comparison of the analytical solution with the computed results, the accuracy and stability of this improved hybrid method are discussed. Satisfactory results are found for both weak and strong diffusion problems under large Courant number conditions. The sensitivity of the improved method to the temporal weighting factor has also been demonstrated. For strong diffusion problems, the use of a larger temporal weighting factor becomes necessary to eliminate the phenomenon of instability.     

Stability properties of the Discontinuous Galerkin Material Point Method for hyperbolic problems in one and two space dimensions

In this paper, stability conditions are derived for the Discontinuous Galerkin Material Point Method (DGMPM) on the scalar linear advection equation for the sake of simplicity and without loss of generality for linear problems. The discrete systems resulting from the application of the DGMPM discretization in one and two space dimensions are first written. For these problems, a second-order Runge-Kutta and the forward Euler time discretizations are respectively considered. Moreover, the numerical fluxes are computed at cell faces by means of either the Donor-Cell Upwind or the Corner Transport Upwind methods for multidimensional problems. Second, the discrete scheme equations are derived assuming that all cells of a background grid contain at least one particle. Although a Cartesian grid is considered in two space dimensions, the results can be extended to regular grids. The von Neumann linear stability analysis then allows the computation of the critical Courant number for a given space discretization. Although the DGMPM is equivalent to the first-order finite volume method if one particle lies in each element, so that the Courant number can be set to unity, other distributions of particles may restrict the stability region of the scheme. The study of several configurations is then proposed.     

A higher-order implicit IDO scheme and its CFD application to local mesh refinement method

The Interpolated Differential Operator (IDO) scheme has been developed for the numerical solution of the fluid motion equations, and allows to produce highly accurate results by introducing the spatial derivative of the physical value as an additional dependent variable. For incompressible flows, semi-implicit time integration is strongly affected by the Courant and diffusion number limitation. A high-order fully-implicit IDO scheme is presented, and the two-stage implicit Runge-Kutta time integration keeps over third-order accuracy. The application of the method to the direct numerical simulation of turbulence demonstrates that the proposed scheme retains a resolution comparable to that of spectral methods even for relatively large Courant numbers. The scheme is further applied to the Local Mesh Refinement (LMR) method, where the size of the time step is often restricted by the dimension of the smallest meshes. In the computation of the Karman vortex street problem, the implicit IDO scheme with LMR is shown to allow a conspicuous saving of computational resources.     

A study of domain decomposition and parallel computation

Summary The domain decomposition method and the patched composite grid technique are combined to simulate the flow field around a complex geometry configuration. In this approach the coupling condition on the interfaces between subdomains is important. Two coupling calculation algorithms on interfaces are used in the computation for regular and irregular patched grids. A parallel computing technique on workstation- or PC-network in PVM environment is used in the flow simulation. Numerical results show that the present method is an effective and powerful analysis tool to predict the flow field around a complex geometry.     

Failures of local approximation in finite-element methods

It is shown that standard finite-element discretizations of second-order differential equations (i.e. Galerkin and subdomain methods) using conforming linear elements may fail to approximate the original equation locally if the finite-element grid is irregular or if subdomains are chosen improperly. This failure of local approximation can lead to spurious computational results when subdomain methods are used, but these difficulties can be averted by a judicious choice of subdomains. The conditions which the subdomains must satisfy in order for local approximation to hold are derived and used to construct an algorithm for choosing them properly. The relation of these local results to the global convergence properties of the Galerkin method is discussed.     

Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity

Overlapping Schwarz methods are considered for mixed finite element approximations of linear elasticity, with discontinuous pressure spaces, as well as for compressible elasticity approximated by standard conforming finite elements. The coarse components of the preconditioners are based on spaces, with a number of degrees of freedom per subdomain which are uniformly bounded, which are similar to those previously developed for scalar elliptic problems and domain decomposition methods of iterative substructuring type, i.e. methods based on nonoverlapping decompositions of the domain. The local components of the new preconditioners are based on solvers on a set of overlapping subdomains. In the current study, the dimension of the coarse spaces is smaller than in recently developed algorithms; in the compressible case all independent face degrees of freedom have been eliminated while in the almost incompressible case five out of six are not needed. In many cases, this will result in a reduction of the dimension of the coarse space by about one half compared with that of the algorithm previously considered. In addition, in spite of using overlapping subdomains to define the local components of the preconditioner, values of the residual and the approximate solution need only to be retained on the interface between the subdomains in the iteration of the new hybrid Schwarz algorithm. The use of discontinuous pressures makes it possible to work exclusively with symmetric, positive‐definite problems and the standard preconditioned conjugate gradient method. Bounds are established for the condition number of the preconditioned operators. The bound for the almost incompressible case grows in proportion to the square of the logarithm of the number of degrees of freedom of individual subdomains and the third power of the relative overlap between the overlapping subdomains, and it is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. Numerical results illustrate the findings. Copyright © 2009 John Wiley & Sons, Ltd.     

AN ADAPTIVE SUCCESSIVE OVER RELAXATION DOMAIN DECOMPOSITION BY THE BOUNDARY SPECTRAL STRIP METHOD

In the present paper a new adaptive successive over relaxation domain decomposition technique is developed for the boundary spectral strip method. The proposed scheme is based on dividing the overall domain of the problem into several subdomains. First each of the subdomains in the BIEM matrices is analysed independently. These matrices together with an arbitrary initial guess of displacements on the interface of each two neighbouring subdomains, enable an iterative and a very efficient solution of the whole problem. An adaptive procedure, based on comparing two norms along the interface of subregions, is carried out to impose successive over relaxation convergence. Numerical results comparing the present scheme with single domain solutions emphasize the capability of the proposed technique regarding accuracy and computational efforts. © 1997 by John Wiley & Sons, Ltd.     

Source terms in Eulerian–Lagrangian contaminant transport simulation

Standard Eulerian treatment of source terms in Eulerian–Lagrangian numerical simulations results in poor performance at higher Courant numbers. To regain the customary high accuracy of Eulerian–Lagrangian methods under these conditions, a Lagrangian treatment of source terms is needed. It is also important to include the effects of fluid sources as well as contaminant sources. A new Lagrangian source formulation is presented, which has been implemented in a finite element simulator for contaminant transport in rivers and estuaries. Test problems demonstrate the high accuracy of the technique under a range of conditions, and its applicability to general multi‐dimensional problems and unstructured grids. Copyright © 2004 John Wiley & Sons, Ltd.     

A mesh free approach using radial basis functions and parallel domain decomposition for solving three‐dimensional diffusion equations

The analysis of transient heat conduction problems in large, complex computational domains is a problem of interest in many technological applications including electronic cooling, encapsulation using functionally graded composite materials, and cryogenics. In many of these applications, the domains may be multiply connected and contain moving boundaries making it desirable to consider meshless methods of analysis. The method of fundamental solutions along with a parallel domain decomposition method is developed for the solution of three‐dimensional parabolic differential equations. In the current approach, time is discretized using the generalized trapezoidal rule transforming the original parabolic partial differential equation into a sequence of non‐homogeneous modified Helmholtz equations. An approximate particular solution is derived using polyharmonic splines. Interfacial conditions between subdomains are satisfied using a Schwarz Neumann–Neumann iteration scheme. Outside of the first time step where zero initial flux is assumed, the initial estimates for the interfacial flux is given from the converged solution obtained during the previous time step. This significantly reduces the number of iterations required to meet the convergence criterion. The accuracy of the method of fundamental solutions approach is demonstrated through two benchmark problems. The parallel efficiency of the domain decomposition method is evaluated by considering cases with 8, 27, and 64 subdomains. Copyright 2004 © John Wiley & Sons, Ltd.     

An implicit multi-time step integration method for structural dynamics problems

A multi-time step integration algorithm is developed based on the trapezoidal rule time integration method for finite element equations of motion. This algorithm uses nodal groups to partition the mesh into subdomains that are updated with different time steps. A␣stability analysis of the method shows that the scheme retains the unconditionally stable behavior of the trapezoidal rule and conserves the same pseudo energy as the parent algorithm. Several numerical examples are used to verify the stability of the method and to investigate the accuracy of the scheme.