Martin Schanz

Poroelastodynamics: Linear Models, Analytical Solutions, and Numerical Methods

This article presents an overview on poroelastodynamic models and some analytical solutions. A brief summary of Biot’s theory and on other poroelastic dynamic governing equations is given. There is a focus on dynamic formulations and the quasi-static case is not considered at all. Some analytical solutions for special problems, fundamental solutions, and Green’s functions are discussed. The numerical realization with two different methodologies, namely the Finite Element Method and the Boundary Element Method, is reviewed. Preprint No 3/2008 Institute of Applied Mechanics

Application of 'Operational Quadrature Methods' in Time Domain Boundary Element Methods

The usual time domain Boundary Element Method (BEM) contains fundamentalsolutions which are convoluted with time-dependent boundary data andintegrated over the boundary surface. Here, a new approach for theevaluation of the convolution integrals, the so-called ’OperationalQuadrature Methods‘ developed by Lubich, is presented. In thisformulation, the convolution integral is numerically approximated by aquadrature formula whose weights are determined using the Laplacetransform of the fundamental solution and a linear multisep method. Tostudy the behaviour of the method, the numerical convolution of afundamental solution with a unit step function is compared with theanalytical result. Then, a time domain Boundary Element formulationapplying the ’Operational Quadrature Methods‘ is derived. For thisformulation only the fundamental solutions in Laplace domain arenecessary. The properties of the new formulation are studied with anumerical example.

Transient wave propagation in a one-dimensional poroelastic column

SummaryBiots theory of porous media governs the wave propagation in a porous, elastic solid infiltrated with fluid. In this theory, a second compressional wave, known as the slow wave, has been identified. In this paper, Biots theory is applied to a one-dimensional continuum. Despite the simplicity of the geometry, an exact solution of the full model, and a detailed analysis of the phenomenon, so far have not been achieved. In the present approach, an analytical solution in the Laplace transform domain is obtained showing clearly two compressional waves. For the special case of an inviscid fluid, a closed form exact solution in time domain is obtained using an analytical inverse Laplace transform. For the general case of a viscous fluid, solution in time domain is evaluated using the Convolution Quadrature Method of Lubich. Of all the inverse methods previously investigated, it seems that only the method of Lubich is efficies and stable enough to handle the highly transient cases such as impact and step loadings. Using properties of three widely different real materials, the wave propagating behavior, in terms of stress, pore pressure, displacement, and flux, are examined. Of most interest is the identification of second compressional wave and its sensitivity of material parameters.

Recent Advances and Emerging Applications of the Boundary Element Method

Sponsored by the U.S. National Science Foundation, a workshop on the boundary element method (BEM) was held on the campus of the University of Akron during September 1–3, 2010 (NSF, 2010, “Workshop on the Emerging Applications and Future Directions of the Boundary Element Method,” University of Akron, Ohio, September 1–3). This paper was prepared after this workshop by the organizers and participants based on the presentations and discussions at the workshop. The paper aims to review the major research achievements in the last decade, the current status, and the future directions of the BEM in the next decade. The review starts with a brief introduction to the BEM. Then, new developments in Greens functions, symmetric Galerkin formulations, boundary meshfree methods, and variationally based BEM formulations are reviewed. Next, fast solution methods for efficiently solving the BEM systems of equations, namely, the fast multipole method, the pre-corrected fast Fourier transformation method, and the adaptive cross approximation method are presented. Emerging applications of the BEM in solving microelectromechanical systems, composites, functionally graded materials, fracture mechanics, acoustic, elastic and electromagnetic waves, time-domain problems, and coupled methods are reviewed. Finally, future directions of the BEM as envisioned by the authors for the next five to ten years are discussed. This paper is intended for students, researchers, and engineers who are new in BEM research and wish to have an overview of the field. Technical details of the BEM and related approaches discussed in the review can be found in the Reference section with more than 400 papers cited in this review.

Application of 3D time domain boundary element formulation to wave propagation in poroelastic solids

Abstract The dynamic responses of fluid-saturated semi-infinite porous continua to transient excitations such as seismic waves or ground vibrations are important in the design of soil-structure systems. Biots theory of porous media governs the wave propagation in a porous elastic solid infiltrated with fluid. The significant difference to an elastic solid is the appearance of the so-called slow compressional wave. The most powerful methodology to tackle wave propagation in a semi-infinite homogeneous poroelastic domain is the boundary element method (BEM). To model the dynamic behavior of a poroelastic material in the time domain, the time domain fundamental solution is needed. Such solution however does not exist in closed form. The recently developed ‘convolution quadrature method’, proposed by Lubich, utilizes the existing Laplace transformed fundamental solution and makes it possible to work in the time domain. Hence, applying this quadrature formula to the time dependent boundary integral equation, a time-stepping procedure is obtained based only on the Laplace domain fundamental solution and a linear multistep method. Finally, two examples show both the accuracy of the proposed time-stepping procedure and the appearance of the slow compressional wave, additionally to the other waves known from elastodynamics.

Wave Propagation Problems Treated with Convolution Quadrature and BEM

Boundary element methods for steady state problems have reached a state of maturity in both analysis and efficient implementation and have become an ubiquitous tool in engineering applications. Their time-domain counterparts, however, in particular for wave propagation phenomena, still present many open questions related to the analysis of the numerical methods and their algorithmic implementation. In recent years many exciting results have been achieved in this area. In this review paper, a particular type of methods for treating time-domain boundary integral equations (TDBIE), the convolution quadrature, is described together with application areas and most recent improvements to the analysis and efficient implementation. An important attraction of these methods is their intrinsic stability, often a problem with numerical methods for TDBIE of wave propagation. Further, since convolution quadrature, though a time-domain method, uses only the kernel of the integral operator in the Laplace domain, it is widely applicable also to problems such as viscoelastodynamics, where the kernel is known only in the Laplace domain. This makes convolution quadrature for TDBIE an important numerical method for wave propagation problems, which requires further attention.

Fast directional multilevel summation for oscillatory kernels based on Chebyshev interpolation

Many applications lead to large systems of linear equations with dense matrices. Direct matrix-vector products become prohibitive, since the computational cost increases quadratically with the size of the problem. By exploiting specific kernel properties fast algorithms can be constructed. A directional multilevel algorithm for translation-invariant oscillatory kernels of the type K(x,y)=G(x-y)e^i^k^|^x^-^y^|, with G(x-y) being any smooth kernel, will be presented. We will first present a general approach to build fast multipole methods (FMMs) based on Chebyshev interpolation and the adaptive cross approximation (ACA) for smooth kernels. The Chebyshev interpolation is used to transfer information up and down the levels of the FMM. The scheme is further accelerated by compressing the information stored at Chebyshev interpolation points using ACA and QR decompositions. This leads to a nearly optimal computational cost with a small pre-processing time due to the low computational cost of ACA. This approach is in particular faster than performing singular value decompositions. This does not address the difficulties associated with the oscillatory nature of K. For that purpose, we consider the following modification of the kernel K^u=K(x,y)e^-^i^k^u^.^(^x^-^y^), where u is a unit vector (see Brandt [1]). We proved that the kernel K^u can be interpolated efficiently when x-y lies in a cone of direction u. This result is used to construct an FMM for the kernel K. Theoretical error bounds will be presented to control the error in the computation as well as the computational cost of the method. The paper ends with the presentation of 2D and 3D numerical convergence studies, and computational cost benchmarks.

Dynamic analysis of a one-dimensional poroviscoelastic column

The response due to a dynamic loading of a poroviscoelastic one-dimensional column is treated analytically. Biots theory of poroelasticity is generalized to poroviscoelasticity using the elastic-viscoelastic correspondence principle in the Laplace domain. Damping effects of the solid skeletal structure and the solid material itself are taken into account. The fluid is modeled as in the original Biots theory without any viscoelastic effects. The solution of the governing set of two coupled differential equations known from the purely poroelastic case is converted to the poroviscoelastic solution using the developed elasticviscoelastic correspondence in Laplace domain. The time-dependent response of the column is achieved by the Convolution Quadrature Method proposed by Lubich. Some interesting effects of viscoelasticity on the response of the column caused by a stress, pressure, and displacement loading are studied.

Boundary Element Calculation of Transient Response of Viscoelastic Solids Based on Inverse Transformation

Mixed boundary value problems of solid mechanics are treated by numericalsolutions of Boundary Integral Equations (BIE) in time domain with theBoundary Element Method (BEM) thus reducing the spatial problem dimensionby one. Viscoelastic constitutive behaviour is implemented by means of aLaplace transform technique based on an elastic--viscoelasticcorrespondence principle. The concept of fractional differintegrationgeneralizes conventional constitutive equations and provides improvedcurve fitting of measured material response with fewer parameters. As theimplementation of viscoelasticity is provided in each time step in theLaplace domain, efficient algorithms for the inverse transformation intime domain are needed. This is why the performance of adapted algorithmsby Talbot, Durbin and Crump are compared. The impact responseof a base plate bonded on a viscoelastic soil halfspace is discussed as anumerical example. Viscous forces increase the velocities of surface wavepropagation and cause attenuation in addition to the so called geometricaldamping by radiation.

A fast Galerkin method for parabolic space-time boundary integral equations

An efficient scheme for solving boundary integral equations of the heat equation based on the Galerkin method is introduced. The parabolic fast multipole method (pFMM) is applied to accelerate the evaluation of the thermal layer potentials. In order to remain attractive for a wide range of applications, a key issue is to ensure efficiency for a big variety of temporal to spatial mesh ratios. Within the parabolic Galerkin FMM (pGFMM) it turns out that the temporal nearfield can become very costly. To that end, a modified fast Gauss transform (FGT) is developed. The complexity and convergence behavior of the method are analyzed and numerically investigated on a range of model problems. The results demonstrate that the complexity is nearly optimal in the number of discretization parameters while the convergence rate of the Galerkin method is preserved.