Showing papers on "Numerical analysis published in 2002"
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29 Jul 2002TL;DR: In this paper, Galerkin et al. defined mesh-free methods for shape function construction, including the use of mesh-less local Petrov-Galerkin methods.
Abstract: Preliminaries Physical Problems in Engineering Solid Mechanics: A Fundamental Engineering Problem Numerical Techniques: Practical Solution Tools Defining Meshfree Methods Need for Meshfree Methods The Ideas of Meshfree Methods Basic Techniques for Meshfree Methods Outline of the Book Some Notations and Default Conventions Remarks Meshfree Shape Function Construction Basic Issues for Shape Function Construction Smoothed Particle Hydrodynamics Approach Reproducing Kernel Particle Method Moving Least Squares Approximation Point Interpolation Method Radial PIM Radial PIM with Polynomial Reproduction Weighted Least Square (WLS) Approximation Polynomial PIM with Rotational Coordinate Transformation Comparison Study via Examples Compatibility Issues: An Analysis Other Methods Function Spaces for Meshfree Methods Function Spaces Useful Spaces in Weak Formulation G Spaces: Definition G1h Spaces: Basic Properties Error Estimation Concluding Remarks Strain Field Construction Why Construct Strain Field? Historical Notes How to Construct? Admissible Conditions for Constructed Strain Fields Strain Construction Techniques Concluding Remarks Weak and Weakened Weak Formulations Introduction to Strong and Weak Forms Weighted Residual Method A Weak Formulation: Galerkin A Weakened Weak Formulation: GS-Galerkin The Hu-Washizu Principle The Hellinger-Reissner Principle The Modified Hellinger-Reissner Principle Single-Field Hellinger-Reissner Principle The Principle of Minimum Complementary Energy The Principle of Minimum Potential Energy Hamilton's Principle Hamilton's Principle with Constraints Galerkin Weak Form Galerkin Weak Form with Constraints A Weakened Weak Formulation: SC-Galerkin Parameterized Mixed Weak Form Concluding Remarks Element Free Galerkin Method EFG Formulation with Lagrange Multipliers EFG with Penalty Method Summary Meshless Local Petrov-Galerkin Method MLPG Formulation MLPG for Dynamic Problems Concluding Remarks Point Interpolation Methods Node-Based Smoothed Point Interpolation Method (NS-PIM) NS-PIM Using Radial Basis Functions (NS-RPIM) Upper Bound Properties of NS-PIM and NS-RPIM Edge-Based Smoothed Point Interpolation Methods (ES-PIMs) A Combined ES/NS Point Interpolation Methods (ES/NS-PIM) Strain-Constructed Point Interpolation Method (SC-PIM) A Comparison Study Summary Meshfree Methods for Fluid Dynamics Problem Introduction Navier-Stokes Equations Smoothed Particle Hydrodynamics Method Gradient Smoothing Method (GSM) Adaptive Gradient Smoothing Method (A-GSM) A Discussion on GSM for Incompressible Flows Other Improvements on GSM Meshfree Methods for Beams PIM Shape Function for Thin Beams Strong Form Equations Weak Formulation: Galerkin Formulation A Weakened Weak Formulation: GS-Galerkin Three Models Formulation for NS-PIM for Thin Beams Formulation for Dynamic Problems Numerical Examples for Static Analysis Numerical Examples: Upper Bound Solution Numerical Examples for Free Vibration Analysis Concluding Remarks Meshfree Methods for Plates Mechanics for Plates EFG Method for Thin Plates EFG Method for Thin Composite Laminates EFG Method for Thick Plates ES-PIM for Plates Meshfree Methods for Shells EFG Method for Spatial Thin Shells EFG Method for Thick Shells ES-PIM for Thick Shells Summary Boundary Meshfree Methods RPIM Using Polynomial Basis RPIM Using Radial Function Basis Remarks Meshfree Methods Coupled with Other Methods Coupled EFG/BEM Coupled EFG and Hybrid BEM Remarks Meshfree Methods for Adaptive Analysis Triangular Mesh and Integration Cells Node Numbering: A Simple Approach Bucket Algorithm for Node Searching Relay Model for Domains with Irregular Boundaries Techniques for Adaptive Analysis Concluding Remarks MFree2D(c) Overview Techniques Used in MFree2D Preprocessing in MFree2D Postprocessing in MFree2D Index References appear at the end of each chapter.
2,121 citations
TL;DR: A new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field that compares favorably with volume of fluid methods in the conservation of mass and purely Lagrangian schemes for interface resolution.
1,238 citations
TL;DR: The method allows for discontinuities, internal to the elements, in the approximation across the interface, and it is shown that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh.
1,013 citations
TL;DR: DDE-BIFTOOL, a Matlab package for numerical bifurcation analysis of systems of delay differential equations with several fixed, discrete delays, is described and its usage and capabilities are illustrated through analysing three examples.
Abstract: We describe DDE-BIFTOOL, a Matlab package for numerical bifurcation analysis of systems of delay differential equations with several fixed, discrete delays. The package implements continuation of steady state solutions and periodic solutions and their stability analysis. It also computes and continues steady state fold and Hopf bifurcations and, from the latter, it can switch to the emanating branch of periodic solutions. We describe the numerical methods upon which the package is based and illustrate its usage and capabilities through analysing three examples: two models of coupled neurons with delayed feedback and a model of two oscillators coupled with delay.
832 citations
TL;DR: The method is ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.
Abstract: We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures, methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage, in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.
684 citations
TL;DR: In this article, the authors present a numerical method for the stability analysis of linear delayed systems based on a special kind of discretization technique with respect to the past effect only.
Abstract: SUMMARY The paper presents an ecient numerical method for the stability analysis of linear delayed systems. The method is based on a special kind of discretization technique with respect to the past eect only. The resulting approximate system is delayed and also time periodic, but still, it can be transformed analytically into a high-dimensional linear discrete system. The method is applied to determine the stability charts of the Mathieu equation with continuous time delay. Copyright ? 2002 John Wiley & Sons, Ltd.
670 citations
TL;DR: In this article, the variable coefficient Poisson equation with Dirichlet boundary conditions on an irregular domain is considered and a symmetric implicit time discretization matrix is proposed to obtain second-order accuracy.
546 citations
TL;DR: A continuous/discontinuous Galerkin (C/DG) method is proposed which uses C0-continuous interpolation functions and is formulated in the primary variable only, leading to a formulation where displacements are the only degrees offreedom, and no rotational degrees of freedom need to be considered.
541 citations
TL;DR: Weaknesses and inconsistencies of current model-verification methods are discussed as well as benchmark solutions for solving the coupled spatio-temporal convection process, consistent velocity approximation, and error-based mesh adaptation techniques.
504 citations
TL;DR: In this paper, an iterative method to compute the solution of Navier-Stokes and shallow water equations for surface flows and Darcy's equation for groundwater flows is proposed.
493 citations
TL;DR: In this paper, a stabilized finite element method is proposed to solve the transient Navier-Stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales.
TL;DR: In this article, a mathematical model for a finite-strain elastoplastic evolution problem is proposed in which one time-step of an implicit time-discretization leads to generally non-convex minimization problems.
Abstract: A mathematical model for a finite–strain elastoplastic evolution problem is proposed in which one time–step of an implicit time–discretization leads to generally non–convex minimization problems. The elimination of all internal variables enables a mathematical and numerical analysis of a reduced problem within the general framework of calculus of variations and nonlinear partial differential equations. The results for a single slip–system and von Mises plasticity illustrate that finite–strain elastoplasticity generates reduced problems with non–quasiconvex energy densities and so allows for non–attainment of energy minimizers and microstructures.
TL;DR: In this paper, a new trigger and multi-cell profiles with four square elements at the corner were developed to improve the crash energy absorption and weight efficiency in terms of energy consumption.
Abstract: New types of trigger and multi-cell profiles with four square elements at the corner are developed. In terms of the crash energy absorption and weight efficiency, the new multi-cell structure shows dramatic improvements over the conventional square box column. The optimization process with the target of maximizing the specific energy absorption has been successfully carried out, and the example of design process is provided. In the optimization process, the problem of stable progressive folding is also addressed. The analytical solution for calculating the mean crushing force of new multi-cell profiles is derived showing good agreement with the numerical results. Finally, the advantage of the new design over the conventional single or multi-cell profiles is discussed.
TL;DR: The scaled boundary finite element method as discussed by the authors is a semi-analytical technique that combines the advantages of the finite element and the boundary element methods with unique properties of its own, such as axisymmetry.
Abstract: The scaled-boundary finite element method is a novel semi-analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. This paper develops a new virtual work formulation and modal interpretation of the method for elastostatics. This formulation follows a similar procedure to the traditional virtual work derivation of the standard finite element method. As well as making the method more accessible, this approach leads to new techniques for the treatment of body loads, side-face loads and axisymmetry that simplify implementation. The paper fully develops the new formulation, and provides four examples illustrating the versatility, accuracy and efficiency of the scaled boundary finite-element method. Both bounded and unbounded domains are treated, together with problems involving stress singularities.
TL;DR: In this article, the authors developed new stabilized mixed finite element methods for Darcy flow and established stability and an a priori error estimate in the stability norm for a wide variety of convergent finite elements.
TL;DR: In this paper, the authors evaluate linear elastic stress fields in the neighborhood of U-and V-shaped notches in plane plates. But the main aim is to improve the accuracy of an approximate solution already proposed in the literature by changing the polynomial arrangement of complex potential functions and properly adapting the boundary conditions.
TL;DR: The ghost fluid method is used to create accurate discretizations across the Eulerian/Lagrangian interface and is presented in both one and two spatial dimensions; three-dimensional extensions (to the interface coupling method) are straightforward.
TL;DR: It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-best N -term approximation.
Abstract: This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet-based method developed in [17] for symmetric positive definite problems to indefinite or unsymmetric systems of operator equations. This is accomplished by first introducing techniques (such as the least squares formulation developed in [26]) that transform the original (continuous) problem into an equivalent infinite system of equations which is now well-posed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve the resulting infinite system of equations. This second step requires a significant modification of the ideas from [17]. The main departure from [17] is to develop an iterative scheme that directly applies to the infinite-dimensional problem rather than finite subproblems derived from the infinite problem. This rests on an adaptive application of the infinite-dimensional operator to finite vectors representing elements from finite-dimensional trial spaces. It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-best N -term approximation. An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of trial spaces, like the LBB condition, no longer arise.
TL;DR: A new method of regularizing time harmonic Maxwell equations by a {\bf grad}-div term adapted to the geometry of the domain is presented, which proves to be numerically efficient.
Abstract: We present a new method of regularizing time harmonic Maxwell equations by a {\bf grad}-div term adapted to the geometry of the domain. This method applies to polygonal domains in two dimensions as well as to polyhedral domains in three dimensions. In the presence of reentrant corners or edges, the usual regularization is known to produce wrong solutions due the non-density of smooth fields in the variational space. We get rid of this undesirable effect by the introduction of special weights inside the divergence integral. Standard finite elements can then be used for the approximation of the solution. This method proves to be numerically efficient.
TL;DR: The accuracy, stability, and reliability of a numerical model based on a Godunov-type scheme are verified, through a comparison between calculated results and observed data for the Malpasset dam-break event, which occurred in southern France in 1959.
Abstract: The accuracy, stability, and reliability of a numerical model based on a Godunov-type scheme are verified in this paper, through a comparison between calculated results and observed data for the Malpasset dam-break event, which occurred in southern France in 1959. This event is an unique opportunity for code validation because of the availability of extensive field data on the flooding wave due to the dam failure. In the code the shallow water equations are discretized using the finite volume method, and the numerical model allows second order accuracy, both in space and time. The classical Godunov approach is used. More specifically, the Harten, Lax, and van Leer Riemann solver is applied. The resulting scheme is of high resolution and satisfies the total variation diminishing condition. For the numerical treatment of source terms relative to the friction slope, a semi-implicit technique is used, while for the source terms relative to the bottom slope a new explicit method is developed and tested.
TL;DR: In this article, the spectral stochastic finite element method (SSFEM) is considered in conjunction with the first-order reliability method (FORM) and with importance sampling for finite element reliability analysis.
TL;DR: In this article, a variational formulation for the homogenization analysis of inelastic solid materials undergoing finite strains is presented, where a quasi-hyperelastic micro-structure micro-stress potential is obtained from a local minimization problem with respect to the internal variables.
Abstract: The paper presents new continuous and discrete variational formulations for the homogenization analysis of inelastic solid materials undergoing finite strains. The point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro-structure as a generalized standard medium in terms of an energy storage and a dissipation function. Consistent with this type of finite inelasticity we develop a new incremental variational formulation of the local constitutive response, where a quasi-hyperelastic micro-stress potential is obtained from a local minimization problem with respect to the internal variables. It is shown that this local minimization problem determines the internal state of the material for finite increments of time. We specify the local variational formulation for a distinct setting of multi-surface inelasticity and develop a numerical solution technique based on a time discretization of the internal variables. The existence of the quasi-hyperelastic stress potential allows the extension of homogenization approaches of finite elasticity to the incremental setting of finite inelasticity. Focussing on macro-deformation-driven micro-structures, we develop a new incremental variational formulation of the global homogenization problem for generalized standard materials at finite strains, where a quasi-hyperelastic macro-stress potential is obtained from a global minimization problem with respect to the fine-scale displacement fluctuation field. It is shown that this global minimization problem determines the state of the micro-structure for finite increments of time. We consider three different settings of the global variational problem for prescribed displacements, non-trivial periodic displacements and prescribed stresses on the boundary of the micro-structure and develop numerical solution methods based on a spatial discretization of the fine-scale displacement fluctuation field. Representative applications of the proposed minimization principles are demonstrated for a constitutive model of crystal plasticity and the homogenization problem of texture analysis in polycrystalline aggregates.
TL;DR: It is demonstrated that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers, which is directly correlated with the convergence properties of iterative solvers.
Abstract: We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers.
TL;DR: In this paper, a new continualization method is proposed in which each higher-order stiffness term is accompanied by a higherorder inertia term, and the resulting models are dynamically consistent.
Abstract: This paper is the first in a series of two that focus on gradient elasticity models derived from a discrete microstructure. In this first paper, a new continualization method is proposed in which each higher-order stiffness term is accompanied by a higher-order inertia term. As such, the resulting models are dynamically consistent. A new parameter is introduced that accounts for the nonlocal interaction between variables of the discrete model and of the continuous model. When this parameter is set to proper values, physically realistic behavior is obtained in statics as well as in dynamics. In this sense, the proposed methodology is superior to earlier approaches to derive gradient elasticity models, in which anomalies in the dynamic behavior have been found. A generic formulation of field equations and boundary conditions is given based on Hamilton's principle. In the second paper, analytical and numerical results of static and dynamic response of the second-order model and the fourth-order model will be treated.
TL;DR: In this article, a new space-time discontinuous Galerkin finite element method for the solution of the Euler equations of gas dynamics in time-dependent flow domains is presented.
TL;DR: In this article, the main techniques, mathematical tools, and existing algorithms for numerical simulation of finite dimensional nonsmooth multibody mechanical systems are reviewed. And the rigid body dynamical case is examined here.
Abstract: This review article focuses on the problems related to numerical simulation of finite dimensional nonsmooth multibody mechanical systems. The rigid body dynamical case is examined here. This class of systems involves complementarity conditions and impact phenomena, which make its study and numerical analysis a difficult problem that cannot be solved by relying on known Ordinary Differential Equation (ODE) or Differential Algebraic Equation (DAE) integrators only. The main techniques, mathematical tools, and existing algorithms are reviewed.
TL;DR: A new active-set method for smooth box-constrained minimization is introduced, which combines an unconstrained method, including a new line-search which aims to add many constraints to the working set at a single iteration, with a recently introduced technique for dropping constraints from theWorking set.
Abstract: A new active-set method for smooth box-constrained minimization is introduced. The algorithm combines an unconstrained method, including a new line-search which aims to add many constraints to the working set at a single iteration, with a recently introduced technique (spectral projected gradient) for dropping constraints from the working set. Global convergence is proved. A computer implementation is fully described and a numerical comparison assesses the reliability of the new algorithm.
TL;DR: In this paper, a rate-independent elastoplastic constitutive model for biological fiber-reinforced composite materials is presented, which is suitable for describing the mechanical behavior of biological fiber reinforced composites in finite elastic and plastic strain domains.
Abstract: This paper presents a rate-independent elastoplastic constitutive model for (nearly) incompressible biological fiber-reinforced composite materials. The constitutive framework, based on multisurface plasticity, is suitable for describing the mechanical behavior of biological fiber-reinforced composites in finite elastic and plastic strain domains. A key point of the constitutive model is the use of slip systems, which determine the strongly anisotropic elastic and plastic behavior of biological fiber-reinforced composites. The multiplicative decomposition of the deformation gradient into elastic and plastic parts allows the introduction of an anisotropic Helmholtz free-energy function for determining the anisotropic response. We use the unconditionally stable backward-Euler method to integrate the flow rule and employ the commonly used elastic predictor/plastic corrector concept to update the plastic variables. This choice is expressed as an Eulerian vector update the Newton's type, which leads to a numerically stable and efficient material model. By means of a representative numerical simulations the performance of the proposed constitutive framework is investigated in detail.
TL;DR: Numerical methods for dissipative particle dynamics, a system of stochastic differential equations for simulating particles interacting pairwise according to a soft potential at constant temperature, are studied.
Abstract: We study numerical methods for dissipative particle dynamics, a system of stochastic differential equations for simulating particles interacting pairwise according to a soft potential at constant temperature where the total momentum is conserved. We introduce splitting methods and examine the behavior of these methods experimentally. The performance of the methods, particularly temperature control, is compared to the modified velocity Verlet method used in many previous papers.