Dynamic response of embedded foundations: A hybrid approach

A hybrid approach is presented to obtain the dynamic response of foundations of arbitrary shape embedded in a layered viscoelastic half-space when subjected to external forces and elastic waves. The approach is based on the use of Green's functions for the half-space continuum combined with a finite element discretization of the finite portion of soil excavated for the foundation. Compared with direct or indirect boundary integral equation methods this particular combination permits a reduction of the number of calculations of Green's functions at the expense of additional finite element calculations.The validity and accuracy of the hybrid approach is investigated by comparison with solutions obtained by other methods for axisymmetric foundations embedded in a uniform viscoelastic half-space and in a layered medium. It is found that the proposed hybrid approach with an appropriate discretization of the soil excavated for the foundation can achieve excellent accuracy.     

A boundary element analysis of the linear elasto-static axisymmetric contact problem of an infinite layered medium with a homogeneous body

A boundary element method for analyzing the linear elasto-static frictionless axisymmetric contact problem of a multi-layered medium with a homogeneous body is presented in this work. In the present method, the boundary element method using Kelvin's fundamental solution is employed to deal with the homogeneous body, while a newly developed boundary integral approach for linear elasto-static axisymmetric problems is utilized to evaluate the distributions of the stresses and displacements of the multi-layered medium in contact status. With the newly developed boundary integral approach for the multi-layered problems, the element discretization is only needed in the contact area of the multi-layered medium regardless of the variations of layer thickness, and the singularities that occur in the common boundary element are thoroughly avoided. Some numerical tests are carried out to show the efficiency of the present method.     

Adaptive FE-procedures in shape optimization

In structural optimization the quality of the optimization result strongly depends on the reliability of the underlying structural analysis. This comprises the quality and range of the mechanical model, e.g. linear elastic or geometrically and materially nonlinear, as well as the accuracy of the numerical model, e.g. the discretization error of the FE-model. The latter aspect is addressed in the present contribution. In order to guarantee the quality of the numerical results the discretization error of the finite element solution is controlled and the finite element discretization is adaptively refined during the optimization process. Conventionally, so-called global error estimates are applied in structural optimization which estimate the error of the total strain energy. In the present paper local error estimates are introduced in shape optimization which allow to control directly the discretization error of local optimization criteria. In general, the adaptive refinement of the finite element discretization by remeshing affects the convergence of the optimization process if a gradient-based optimization algorithm is applied. In order to reduce this effect the sensitivity of the discretization error must also be restricted. Suitable refinement indicators are developed for globally and locally adaptive procedures. Finally, the potential of two techniques, which may improve the numerical efficiency of adaptive FE-procedures within the optimization process, is studied. The proposed methods and procedures are verified by 2-D shape optimization examples. Received June 3, 1999     

A fully discrete scheme for the pressure–stress formulation of the time-domain fluid–structure interaction problem

We propose an implicit Newmark method for the time integration of the pressure–stress formulation of a fluid–structure interaction problem. The space Galerkin discretization is based on the Arnold–Falk–Winther mixed finite element method with weak symmetry in the solid and the usual Lagrange finite element method in the acoustic medium. We prove that the resulting fully discrete scheme is well-posed and uniformly stable with respect to the discretization parameters and Poisson ratio, and we provide asymptotic error estimates. Finally, we present numerical tests to confirm the asymptotic error estimates predicted by the theory.     

On error estimates of the fully discrete penalty method for the viscoelastic flow problem

In this paper, a fully discrete finite element penalty method is presented for the two-dimensional viscoelastic flow problem arising in the Oldroyd model, in which the spatial discretization is based on the finite element approximation and the time discretization is based on the backward Euler scheme. Moreover, we provide the optimal error estimate for the numerical solution under some realistic assumptions. Finally, some numerical experiments are shown to illustrate the efficiency of the penalty method.     

A finite element penalty method for the linearized viscoelastic Oldroyd fluid motion equations

In this paper, a fully discrete finite element penalty method is considered for the two-dimensional linearized viscoelastic fluid motion equations, arising from the Oldroyd model for the non-Newton fluid flows. With the finite element method for the spatial discretization and the backward Euler scheme for the temporal discretization, the velocity and pressure are decoupled in this method, which leads to a large reduction of the computational scale. Under some realistic assumptions, the unconditional stability of the fully discrete scheme is proved. Moreover, the optimal error estimates are obtained, which are better than the existing results. Finally, some numerical results are given to verify the theoretical analysis. The difference between the motion of the Newton and non-Newton fluid is also observed.     

Post-processing and visualization techniques in 2D boundary element analysis

Most post-processors for boundary element (BE) analysis use an auxiliary domain mesh to display domain results, working against the profitable modelling process of a pure boundary discretization. This paper introduces a novel visualization technique which preserves the basic properties of the boundary element methods. The proposed algorithm does not require any domain discretization and is based on the direct and automatic identification of isolines. Another critical aspect of the visualization of domain results in BE analysis is the effort required to evaluate results in interior points. In order to tackle this issue, the present article also provides a comparison between the performance of two different BE formulations (conventional and hybrid). In addition, this paper presents an overview of the most common post-processing and visualization techniques in BE analysis, such as the classical algorithms of scan line and the interpolation over a domain discretization. The results presented herein show that the proposed algorithm offers a very high performance compared with other visualization procedures.     

An Algebraic Multigrid Method for Higher-order Finite Element Discretizations

In this paper, we will design and analyze a class of new algebraic multigrid methods for algebraic systems arising from the discretization of second order elliptic boundary value problems by high-order finite element methods. For a given sparse stiffness matrix from a quadratic or cubic Lagrangian finite element discretization, an algebraic approach is carefully designed to recover the stiffness matrix associated with the linear finite element disretization on the same underlying (but nevertheless unknown to the user) finite element grid. With any given classical algebraic multigrid solver for linear finite element stiffness matrix, a corresponding algebraic multigrid method can then be designed for the quadratic or higher order finite element stiffness matrix by combining with a standard smoother for the original system. This method is designed under the assumption that the sparse matrix to be solved is associated with a specific higher order, quadratic for example, finite element discretization on a finite element grid but the geometric data for the underlying grid is unknown. The resulting new algebraic multigrid method is shown, by numerical experiments, to be much more efficient than the classical algebraic multigrid method which is directly applied to the high-order finite element matrix. Some theoretical analysis is also provided for the convergence of the new method.     

Stability and convergence of the numerical solution for the species equation of a model for PEMFCs

Many mathematical models for fuel cells are presented in the literature, however, few studies of numerical analysis of these models are found. The novelty of this work is the proof of stability and convergence of the equation of chemical species, providing greater reliability to the mathematical model. In this work, a mathematical model is proposed that calculates the flow and concentration of species for proton exchange membrane fuel cells using the finite element method for spatial discretization and the Crank–Nicolson method for temporal discretization. The model takes into account the losses overpotentials at the anode and at the cathode. The numerical results confirm the proof of convergence which is demonstrated in this work. Finally, a numerical result is shown for the fuel cell power relative to current density, and the result has good relation with the experimental data.     

A computational paradigm for multiresolution topology optimization (MTOP)

This paper presents a multiresolution topology optimization (MTOP) scheme to obtain high resolution designs with relatively low computational cost. We employ three distinct discretization levels for the topology optimization procedure: the displacement mesh (or finite element mesh) to perform the analysis, the design variable mesh to perform the optimization, and the density mesh (or density element mesh) to represent material distribution and compute the stiffness matrices. We employ a coarser discretization for finite elements and finer discretization for both density elements and design variables. A projection scheme is employed to compute the element densities from design variables and control the length scale of the material density. We demonstrate via various two- and three-dimensional numerical examples that the resolution of the design can be significantly improved without refining the finite element mesh.     

An Efficient Discretization of the Navier–Stokes Equations in an Axisymmetric Domain. Part 1: The Discrete Problem and its Numerical Analysis

Any solution of the Navier–Stokes equations in a three-dimensional axisymmetric domain admits a Fourier expansion with respect to the angular variable, and it can be noted that each Fourier coefficient satisfies a variational problem on the meridian domain, all problems being coupled due to the nonlinear convection term. We propose a discretization of these equations which combines Fourier truncation and finite element methods applied to each two-dimensional system. We perform the a priori and a posteriori analysis of this discretization.     

A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem

This work is devoted to a posteriori error analysis of fully discrete finite element approximations to the time dependent Stokes system. The space discretization is based on popular stable spaces, including Crouzeix–Raviart and Taylor–Hood finite element methods. Implicit Euler is applied for the time discretization. The finite element spaces are allowed to change with time steps and the projection steps include alternatives that is hoped to cope with possible numerical artifices and the loss of the discrete incompressibility of the schemes. The final estimates are of optimal order in \(L^\infty (L^2) \) for the velocity error.     

Robust multigrid preconditioners for cell-centered finite volume discretization of the high-contrast diffusion equation

We study a conservative 5-point cell-centered finite volume discretization of the high-contrast diffusion equation. We aim to construct preconditioners that are robust with respect to the magnitude of the coefficient contrast and the mesh size simultaneously. For that, we prove and numerically demonstrate the robustness of the preconditioner proposed by Aksoylu et al. (Comput Vis Sci 11:319–331, 2008) by extending the devised singular perturbation analysis from linear finite element discretization to the above discretization. The singular perturbation analysis is more involved than that of finite element case because all the subblocks in the discretization matrix depend on the diffusion coefficient. However, as the diffusion coefficient approaches infinity, that dependence is eliminated. This allows the same preconditioner to be utilized due to similar limiting behaviours of the submatrices; leading to a narrowing family of preconditioners that can be used for different discretizations. Therefore, we have accomplished a desirable preconditioner design goal. We compare our numerical results to standard cell-centered multigrid implementations and observe that performance of our preconditioner is independent of the utilized smoothers and prolongation operators. As a side result, we also prove a fundamental qualitative property of solution of the high-contrast diffusion equation. Namely, the solution over the highly-diffusive island becomes constant asymptotically. Integration of this qualitative understanding of the underlying PDE to our preconditioner is the main reason behind its superior performance. Diagonal scaling is probably the most basic preconditioner for high-contrast coefficients. Extending the matrix entry based spectral analysis introduced by Graham and Hagger, we rigorously show that the number of small eigenvalues of the diagonally scaled matrix depends on the number of isolated islands comprising the highly-diffusive region. This indicates that diagonal scaling creates a significant clustering of the spectrum, a favorable property for faster convergence of Krylov subspace solvers.     

An error analysis approach for laminated composite plate finite element models

Errors in laminated composite plate finite element models occur at both the individual element level and at the discretization level. This paper shows that parasitic shear causes individual element errors and that its sources must be eliminated if numerically and physically correct results are to be provided by the finite element analysis. In addition, discretization errors occur when the behavior of the continuum is represented by a finite number of degrees of freedom. A procedure to estimate discretization errors in laminated composite plate finite element models and guide refinement, in order to achieve an acceptable level of accuracy, is developed. The error estimator built is based on the energy norm of the error in stress resultants.     

Numerical analysis and testing of a stable and convergent finite element scheme for approximate deconvolution turbulence models

This report presents a stable and convergent finite element scheme for the approximate deconvolution turbulence models (ADM). The ADM is a popular turbulence model intensely studied lately but the computation of its numerical solution raises issues in terms of efficiency and accuracy. This report addresses this question. The proposed scheme presented herein is based on a new interpretation of the ADM model recently introduced by the author. Following this interpretation, the solution of the ADM is viewed as the average of a perturbed Navier–Stokes system. The scheme uses the Crank–Nicolson time discretization and the finite element spatial discretization and is proved to be stable and convergent provided a moderate choice of the time step is made. Numerical tests to verify the convergence rates and performance on a benchmark problem are also provided and they prove the correctness of this approach to numerically solve the ADM.     

Preconditioned Descent Algorithms for p-Laplacian

In this paper, we examine some computational issues on finite element discretization of the p-Laplacian. We introduced a class of descent methods with multi-grid finite element preconditioners, and carried out convergence analysis. We showed that their convergence rate is mesh-independent. We studied the behavior of the algorithms with large p. Our numerical tests show that these algorithms are able to solve large scale p-Laplacian with very large p. The algorithms are then used to solve a variational inequality.      

A high-continuity finite element model for two-dimensional elastic problems

A finite element model for the analysis of two-dimensional elastic problems is presented. The proposed discretization is based on a biquadratic interpolation for the displacement components and takes advantage of the enforcement of the interelement continuity to obtain a profitable reduction of the total number of the degrees of freedom. One node (two kinematical parameters) per element only is required.Numerical results obtained for some test problems show the accuracy of the model in analyzing both the deformations and the stress distribution.     

A Least-Squares Spectral Element Formulation for the Stokes Problem

Least-squares spectral element methods seem very promising since they combine the generality of finite element methods with the accuracy of the spectral methods and also the theoretical and computational advantages in the algorithmic design and implementation of the least-squares methods. The new element in this work is the choice of spectral elements for the discretization of the least-squares formulation for its superior accuracy due to the high-order basis-functions. The main issue of this paper is the derivation of a least-squares spectral element formulation for the Stokes equations and the role of the boundary conditions on the coercivity relations. The numerical simulations confirm the usual exponential rate of convergence when p-refinement is applied which is typical for spectral element discretization.     

On some current procedures and difficulties in finite element analysis of elastic-plastic response

Some finite element procedures for the analysis of elastic-plastic response are presented and critically discussed: a consistent large displacement and large strain formulation is summarized, a versatile elastic-plastic model applicable to the analysis of metals and geological materials is presented, the choice of an appropriate finite element discretization is discussed, and some effective methods for the solution of the nonlinear finite element equations are briefly summarized. Finally, to illustrate the strength and shortcomings of the procedures used, the results of some sample analyses are presented and evaluated.     

双水平井电磁测距径向距离计算方法的ANSYS仿真研究

基于旋转磁场测距(RMRS)的基本理论,借助ANSYS有限元分析软件,研究了金属套管条件下双水平井电磁测距径向距离计算方法。首先建立SAGD双水平井RMRS井下传播模型,通过设定单元属性、划分网格、加载边界条件等对模型进行求解;然后研究套管厚度、直径、相对磁导率等对双水平电磁测距系统中磁场轴向分量的影响;最后利用实验室现有的旋转磁场测距模拟装置对仿真结果进行了验证。研究结果表明：随着套管厚度、直径、相对磁导率的增加,探管接收到的磁感应强度会逐渐减小,但双水平井径向间距计算仍可采用均匀介质中的理论测距导向计算方法。此结论可为套管的选取、磁导向仪器的研究及测量资料解释提供理论参考。