Showing papers on "Discretization published in 1999"
TL;DR: In this paper, a three-dimensional finite deformation cohesive element and a class of irreversible cohesive laws are proposed to track dynamic growing cracks in a drop-weight dynamic fracture test.
Abstract: SUMMARY We develop a three-dimensional nite-deformation cohesive element and a class of irreversible cohesive laws which enable the accurate and ecient tracking of dynamically growing cracks. The cohesive element governs the separation of the crack anks in accordance with an irreversible cohesive law, eventually leading to the formation of free surfaces, and is compatible with a conventional nite element discretization of the bulk material. The versatility and predictive ability of the method is demonstrated through the simulation of a drop-weight dynamic fracture test similar to those reported by Zehnder and Rosakis. 1 The ability of the method to approximate the experimentally observed crack-tip trajectory is particularly noteworthy. Copyright ? 1999 John Wiley & Sons, Ltd.
1,375 citations
TL;DR: In this article, the spectral element method is used for the calculation of synthetic seismograms in 3D earth models using a weak formulation of the equations of motion, which are solved on a mesh of hexahedral elements.
Abstract: SUMMARY We present an introduction to the spectral element method, which provides an innovative numerical approach to the calculation of synthetic seismograms in 3-D earth models. The method combines the £exibility of a ¢nite element method with the accuracy of a spectral method. One uses a weak formulation of the equations of motion, which are solved on a mesh of hexahedral elements that is adapted to the free surface and to the main internal discontinuities of the model. The wave¢eld on the elements is discretized using high-degree Lagrange interpolants, and integration over an element is accomplished based upon the Gauss^Lobatto^Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix, which greatly simpli¢es the algorithm. We illustrate the great potential of the method by comparing it to a discrete wavenumber/re£ectivity method for layer-cake models. Both body and surface waves are accurately represented, and the method can handle point force as well as moment tensor sources. For a model with very steep surface topography we successfully benchmark the method against an approximate boundary technique. For a homogeneous medium with strong attenuation we obtain excellent agreement with the analytical solution for a point force.
1,184 citations
TL;DR: In this article, a new Lagrange-multiplier based fictitious-domain method is presented for the direct numerical simulation of viscous incompressible flow with suspended solid particles, which uses a finite-element discretization in space and an operator-splitting technique for discretisation in time.
1,072 citations
TL;DR: A new model and a solution method for two-phase compressible flows is proposed that provides reliable results, is able to compute strong shock waves, and deals with complex equations of state.
906 citations
TL;DR: A Cartesian grid method has been developed for simulating two-dimensional unsteady, viscous, incompressible flows with complex immersed boundaries and the ability of the solver to simulate flows with very complicated immersed boundaries is demonstrated.
811 citations
TL;DR: The approximation problem, issued from a discretization of a second order elliptic equation in 2D, is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms.
Abstract: The present paper deals with a variant of a non conforming domain decomposition technique: the mortar finite element method. In the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices (and the edges in 3D) of subdomains. It is shown that, written under primal hybrid formulation, the approximation problem, issued from a discretization of a second order elliptic equation in 2D, is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms. Finally the parallelization advantages consequence of this variant are also addressed.
705 citations
TL;DR: In this paper, the deformation of a micro-structure is coupled with the local deformation at a typical material point of the macro-continuum by three alternative constraints of the microscopic fluctuation field.
594 citations
TL;DR: In this article, the authors present efficient techniques for the numerical approximation of complicated dynamical behavior, which allow them to approximate Sinai-Ruelle-Bowen (SRB)-measures as well as (almost) cyclic behavior of a dynamical system.
Abstract: We present efficient techniques for the numerical approximation of complicated dynamical behavior. In particular, we develop numerical methods which allow us to approximate Sinai--Ruelle--Bowen (SRB)-measures as well as (almost) cyclic behavior of a dynamical system. The methods are based on an appropriate discretization of the Frobenius--Perron operator, and two essentially different mathematical concepts are used: our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of SRB-measures by invariant measures of stochastically perturbed systems. The efficiency of the methods is illustrated by several numerical examples.
577 citations
01 Jul 1999
TL;DR: An algorithm for fast, physically accurate simulation of deformable objects suitable for real time animation and virtual environment interaction and how to exploit the coherence of typical interactions to achieve low latency is presented.
Abstract: We present an algorithm for fast, physically accurate simulation of deformable objects suitable for real time animation and virtual environment interaction. We describe the boundary integral equation formulation of static linear elasticity as well as the related Boundary Element Method (BEM) discretization technique. In addition, we show how to exploit the coherence of typical interactions to achieve low latency; the boundary formulation lends itself well to a fast update method when a few boundary conditions change. The algorithms are described in detail with examples from ArtDefo, our implementation.
561 citations
Book•
04 May 1999
TL;DR: In this paper, the basic Equations and Finite Element Discretization are used for error control in time and space and for validation of prediction by centrifuge, respectively.
Abstract: The Basic Equations. Finite Element Discretization. Error Control in Time and Space. Constitutive Relations. Identification of Model Parameters. Static and Quasi--Static Solution. Validation of Prediction by Centrifuge. Prediction Applications and Back Analysis. Multiphase Examples.
454 citations
TL;DR: In this article, the authors proposed an alternative approach for the computation of essential features of the dynamics of Hamiltonian systems (such as molecular dynamics) based on subdivision techniques applied to the Frobenius?Perron operator for the dynamical system.
TL;DR: POD is utilized to solve open-loop and closed-loop optimal control problems for the Burgers equation to comparison of POD-based algorithms with numerical results obtained from finite-element discretization of the optimality system.
Abstract: Proper orthogonal decomposition (POD) is a method to derive reduced-order models for dynamical systems In this paper, POD is utilized to solve open-loop and closed-loop optimal control problems for the Burgers equation The relative simplicity of the equation allows comparison of POD-based algorithms with numerical results obtained from finite-element discretization of the optimality system For closed-loop control, suboptimal state feedback strategies are presented
TL;DR: In this paper, a monotonically integrated large eddy simulation (MILES) approach is proposed, which involves solving the unfiltered Navier-Stokes equations (NSEs) using high-resolution monotone algorithms.
Abstract: With a view to ensure that proper interaction between resolvable or grid scale and subgrid scale (SGS) motions are mimicked, it is vital to determine the necessary physics that must be built into the SGS models. In ordinary large eddy simulation (LES) approaches, models are introduced for closure in the low-pass filtered Navier-Stokes equations (NSEs), which are the ones solved numerically. A promising LES approach is monotonically integrated LES (MILES), which involves solving the unfiltered NSE using high-resolution monotone algorithms; in this approach, implicit SGS models, provided by intrinsic nonlinear high-frequency filters built into the convection discretization, are coupled naturally to the resolvable scales of the flow. Formal properties of the effectual SGS modeling using MILES are documented using databases of simulated homogeneous turbulence and transitional freejets; mathematical and physical aspects of (implicit) SGS modeling through the use of nonlinear flux limiters are addressed in this context
TL;DR: In this article, a computational analysis of hygro-thermal and mechanical behavior of concrete structures at high temperature is presented, and the evaluation of thermal, hygral and mechanical performance of this material, including damage effects, needs the knowledge of the heat and mass transfer processes.
Abstract: A computational analysis of hygro-thermal and mechanical behaviour of concrete structures at high temperature is presented. The evaluation of thermal, hygral and mechanical performance of this material, including damage effects, needs the knowledge of the heat and mass transfer processes. These are simulated within the framework of a coupled model where non-linearities due to high temperatures are accounted for. The constitutive equations are discussed in some detail. The discretization of the governing equations is carried out by Finite Elements in space and Finite Differences in time. Copyright © 1999 John Wiley & Sons, Ltd.
TL;DR: In this paper, the issue of discretization of the strain configuration relationships in the geometrically exact theory of 3D beams has been discussed, which has been at the heart of most recent work.
Abstract: The paper discusses the issue of discretization of the strainconfiguration relationships in the geometrically exact theory of threedimensional (3D) beams, which has been at the heart of most recent...
TL;DR: In this article, a general and readily applicable scheme is presented for the determination of the basis functions that allow the decomposition of the surface current into a solenoidal part and a nonsolenoidal remainder.
Abstract: A general and readily applicable scheme is presented for the determination of the basis functions that allow the decomposition of the surface current into a solenoidal part and a nonsolenoidal remainder. The proposed approach brings into correspondence these two parts with two scalar functions and generates the known loop and star basis functions. The completeness of the loop-star basis is discussed, employing the presented scheme; the issue of the irrotational property of the nonsolenoidal functions is addressed.
TL;DR: In this article, detailed simulation results for the test case of a locally aerated flat bubble column for laminar and turbulent models of Euler-Euler type in two and three dimensions are presented for different space resolutions.
TL;DR: A special admissibility condition for neural activation functions is introduced which requires that the neural activation function be oscillatory and linear transforms are constructed which represent quite general functions f as a superposition of ridge functions.
01 Jan 1999
TL;DR: A meshless approach to the analysis of arbitrary Kirchho plates by the Element-Free Galerkin (EFG) method is presented in this article, which is based on moving least squares approximant.
Abstract: A meshless approach to the analysis of arbitrary Kirchho plates by the Element-Free Galerkin (EFG) method is presented. The method is based on moving least squares approximant. The method is meshless, which means that the discretization is independent of the geometric subdivision into
ite elements". The satisfaction of the C 1 continuity requirements are easily met by EFG since it requires only C 1 weights; therefore, it is not necessary to resort to Mindlin-Reissner theory or to devices such as discrete Kirchho theory. The requirements of consistency are met by the use of a quadratic polynomial basis. A subdivision similar to nite elements is used to provide a background mesh for numerical integration. The essential boundary conditions are enforced by Lagrange multipliers. It is shown, that high accuracy can be achieved for arbitrary grid geometries, for clamped and simply-supported edge conditions, and for regular and irregular grids. Numerical studies are presented which show that the optimal support is about 3:9 node spacings, and that high-order quadrature is required.
TL;DR: In this article, a numerical procedure for the computation of the overall moduli of polycrystalline materials based on a direct evaluation of a micro-macro transition is presented, where the deformation of the micro-structure is coupled with the local deformation at a typical point on the macrocontinuum by three alternative constraints of the microscopic fluctuation field.
TL;DR: A preconditioner for the linearized Navier--Stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero is introduced and it is demonstrated empirically that convergence depends only mildly on these parameters.
Abstract: We introduce a preconditioner for the linearized Navier--Stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero. For constant coefficient problems with periodic boundary conditions, we show that the preconditioning yields a system with a single eigenvalue equal to 1, so that performance is independent of both viscosity and mesh size. For other boundary conditions, we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon. We also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration, with virtually no degradation of performance.
TL;DR: In this article, a class of cell centered finite volume schemes for a linear convection-diusion problem is studied, where the convection and the diusion are respectively approximated by means of an upwind scheme and the so-called diamond cell method.
Abstract: In this paper, a class of cell centered nite volume schemes, on general unstructured meshes, for a linear convection-diusion problem, is studied. The convection and the diusion are respectively approximated by means of an upwind scheme and the so called diamond cell method (4). Our main result is an error estimate of order h, assuming only the W 2;p (for p> 2) regularity of the continuous solution, on a mesh of quadrangles. The proof is based on an extension of the ideas developed in (12). Some new diculties arise here, due to the weak regularity of the solution, and the necessity to approximate the entire gradient, and not only its normal component, as in (12).
TL;DR: In this article, a three-dimensional cohesive element and a class of irreversible cohesive laws were developed to enable the accurate and efficient tracking of three dimensional fatigue crack fronts and the calculation of the attendant fatigue life curves.
TL;DR: New methods for solving nonsymmetric linear systems of equations with multiple right-hand sides based on global oblique and orthogonal projections of the initial matrix residual onto a matrix Krylov subspace are presented.
TL;DR: The central issue is to construct wavelet bases with the desired properties on manifolds which can be represented as the disjoint union of smooth parametric images of the standard cube.
Abstract: The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted seq...
TL;DR: The semi-inversion method as mentioned in this paper is a family of methods based on conversion of a first-kind or strongly-singular second-kind integral equation to a second kind integral equation with a smoother kernel, to ensure pointwise convergence of the usual discretization schemes.
Abstract: We discuss the foundations and state-of-the-art of the method of analytical regularization (MAR) (also called the semi-inversion method). This is a collective name for a family of methods based on conversion of a first-kind or strongly-singular second-kind integral equation to a second-kind integral equation with a smoother kernel, to ensure point-wise convergence of the usual discretization schemes. This is done using analytical inversion of a singular part of the original equation; discretization and semi-inversion can be combined in one operation. Numerous problems being solved today with this approach are reviewed, although in some of them, MAR comes in disguise.
TL;DR: In this article, the authors consider the question of existence and uniqueness of solutions to the spatially homogeneous Boltzmann equation and show that to any initial data with finite mass and energy, there exists a unique solution for which the same two quantities are conserved.
Abstract: We consider the question of existence and uniqueness of solutions to the spatially homogeneous Boltzmann equation. The main result is that to any initial data with finite mass and energy, there exists a unique solution for which the same two quantities are conserved. We also prove that any solution which satisfies certain bounds on moments of order s A second part of the paper is devoted to the time discretization of the Boltzmann equation, the main results being estimates of the rate of convergence for the explicit and implicit Euler schemes. Two auxiliary results are of independent interest: a sharpened form of the so called Povzner inequality, and a regularity result for an iterated gain term.
TL;DR: In this article, a spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two-and three-dimensional geometries.
Abstract: A spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two- and three-dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, defined with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor-product of a Gauss–Lobatto–Legendre 1-D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor/multicorrector format. Long term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two-dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three-dimensional model. Very accurate modelling of Rayleigh wave propagation and surface diffraction is obtained at a low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three-dimensional surface topographies. Copyright © 1999 John Wiley & Sons, Ltd.
TL;DR: It turns out that the characteristic-based-split (CBS) process allows equal interpolation to be used for all system variables without difficulties when the incompressible or nearly incompressable stage is reached.
Abstract: In 1995 the two senior authors of the present paper introduced a new algorithm designed to replace the Taylor–Galerkin (or Lax–Wendroff) methods, used by them so far in the solution of compressible flow problems. The new algorithm was applicable to a wide variety of situations, including fully incompressible flows and shallow water equations, as well as supersonic and hypersonic situations, and has proved to be always at least as accurate as other algorithms currently used. The algorithm is based on the solution of conservation equations of fluid mechanics to avoid any possibility of spurious solutions that may otherwise result. The main aspect of the procedure is to split the equations into two parts, (1) a part that is a set of simple scalar equations of convective–diffusion type for which it is well known that the characteristic Galerkin procedure yields an optimal solution; and (2) the part where the equations are self-adjoint and therefore discretized optimally by the Galerkin procedure. It is possible to solve both the first and second parts of the system explicitly, retaining there the time step limitations of the Taylor–Galerkin procedure. But it is also possible to use semi-implicit processes where in the first part we use a much bigger time step generally governed by the Peclet number of the system while the second part is solved implicitly and is unconditionally stable. It turns out that the characteristic-based-split (CBS) process allows equal interpolation to be used for all system variables without difficulties when the incompressible or nearly incompressible stage is reached. It is hoped that the paper will help to make the algorithm more widely available and understood by the profession and that its advantages can be widely realised. Copyright © 1999 John Wiley & Sons, Ltd.
TL;DR: In this article, a topology optimization method is used to find the distribution of material phases that extremizes an objective function subject to constraints, such as elastic symmetry and volume fractions of the constituent phases, within a periodic base cell.
Abstract: The topology optimization method is used to find the distribution of material phases that extremizes an objective function (e.g., thermal expansion coefficient, piezoelectric coefficients etc) subject to constraints, such as elastic symmetry and volume fractions of the constituent phases, within a periodic base cell. The effective properties of the material structures are found using a numerical homogenization method based on a finite-element discretization of the base cell. The optimization problem is solved using sequential linear programming. We review the topology optimization procedure as a tool for smart materials design and discuss in detail two recent applications of it to design composites with extreme thermal expansion coefficients and piezocomposites with optimal hydrophone characteristics.