Showing papers on "Discretization published in 2010"
TL;DR: A continuous surface charge (CSC) approach is introduced that leads to a smooth and robust formalism for the PCM models and achieves a clear separation between "model" and "cavity" which, together with simple generalizations of modern integral codes, is all that is required for an extensible and efficient implementation of thePCM models.
Abstract: Continuum solvation models are appealing because of the simplified yet accurate description they provide of the solvent effect on a solute, described either by quantum mechanical or classical methods. The polarizable continuum model (PCM) family of solvation models is among the most widely used, although their application has been hampered by discontinuities and singularities arising from the discretization of the integral equations at the solute-solvent interface. In this contribution we introduce a continuous surface charge (CSC) approach that leads to a smooth and robust formalism for the PCM models. We start from the scheme proposed over ten years ago by York and Karplus and we generalize it in various ways, including the extension to analytic second derivatives with respect to atomic positions. We propose an optimal discrete representation of the integral operators required for the determination of the apparent surface charge. We achieve a clear separation between “model” and “cavity” which, together with simple generalizations of modern integral codes, is all that is required for an extensible and efficient implementation of the PCM models. Following this approach we are now able to introduce solvent effects on energies, structures, and vibrational frequencies (analytical first and second derivatives with respect to atomic coordinates), magnetic properties (derivatives with respect of magnetic field using GIAOs), and in the calculation more complex properties like frequency-dependent Raman activities, vibrational circular dichroism, and Raman optical activity.
2,033 citations
TL;DR: In this paper, a Reissner-Mindlin shell formulation based on a degenerated solid is implemented for NURBS-based isogeometric analysis and the performance of the approach is examined on a set of linear elastic and nonlinear elasto-plastic benchmark examples.
621 citations
TL;DR: In this article, the authors consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the well-posedness of the continuous problem.
Abstract: This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the well-posedness of the continuous problem. The discretization methods we consider are finite element methods, in which a variational or weak formulation of the PDE problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace. After a brief introduction to finite element methods, we develop an abstract Hilbert space framework for analyzing the stability and convergence of such discretizations. In this framework, the differential complex is represented by a complex of Hilbert spaces, and stability is obtained by transferring Hodgetheoretic structures that ensure well-posedness of the continuous problem from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they can be arranged into a subcomplex of this Hilbert complex, and there exists a bounded cochain projection from that complex to the subcomplex. In the next part of the paper, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially the elasticity complex and its application to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.
550 citations
TL;DR: High order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for Euler equations of compressible gas dynamics are constructed and extended to higher dimensions on rectangular meshes.
527 citations
TL;DR: An exhaustive study about different discrete-time implementations of resonant controllers, extending to the discretization of the schemes with delay compensation, which is proved to be of great importance in relation with their performance.
Abstract: Resonant controllers have gained significant importance in recent years in multiple applications. Because of their high selectivity, their performance is very dependent on the accuracy of the resonant frequency. An exhaustive study about different discrete-time implementations is contributed in this paper. Some methods, such as the popular ones based on two integrators, cause that the resonant peaks differ from expected. Such inaccuracies result in significant loss of performance, especially for tracking high-frequency signals, since infinite gain at the expected frequency is not achieved, and therefore, zero steady-state error is not assured. Other discretization techniques are demonstrated to be more reliable. The effect on zeros is also analyzed, establishing the influence of each method on the stability. Finally, the study is extended to the discretization of the schemes with delay compensation, which is also proved to be of great importance in relation with their performance. A single-phase active power filter laboratory prototype has been implemented and tested. Experimental results provide a real-time comparison among discretization strategies, which validate the theoretical analysis. The optimum discrete-time implementation alternatives are assessed and summarized.
487 citations
TL;DR: A full-discretization method based on the direct integration scheme for prediction of milling stability based on Floquet theory that has high computational efficiency without loss of any numerical precision is presented.
Abstract: This paper presents a full-discretization method based on the direct integration scheme for prediction of milling stability. The fundamental mathematical model of the dynamic milling process considering the regenerative effect is expressed as a linear time periodic system with a single discrete time delay, and the response of the system is calculated via the direct integration scheme with the help of discretizing the time period. Then, the Duhamel term of the response is solved using the full-discretization method. In each small time interval, the involved system state, time-periodic and time delay items are simultaneously approximated by means of linear interpolation. After obtaining the discrete map of the state transition on one time interval, a closed form expression for the transition matrix of the system is constructed. The milling stability is then predicted based on Floquet theory. The effectiveness of the algorithm is demonstrated by using the benchmark examples for one and two degrees of freedom milling models. It is shown that the proposed method has high computational efficiency without loss of any numerical precision. The code of the algorithm is also attached in the appendix.
435 citations
TL;DR: In this article, a discretisation scheme for heterogeneous anisotropic diffusion problems on general meshes is developed and studied, where the unknowns of this scheme are the values at the center of the control volumes and at some internal interfaces which may for instance be chosen at the diffusion tensor discontinuities.
Abstract: A discretisation scheme for heterogeneous anisotropic diffusion problems on general meshes is developed and studied. The unknowns of this scheme are the values at the centre of the control volumes and at some internal interfaces which may for instance be chosen at the diffusion tensor discontinuities. The scheme is therefore completely cell centred if no edge unknown is kept. It is shown to be accurate on several numerical examples. Mathematical convergence of the approximate solution to the continuous solution is obtained for general (possibly discontinuous) tensors, general (possibly non-conforming) meshes, and with no regularity assumption on the solution. An error estimate is then drawn under sufficient regularity assumptions on the solution.
356 citations
Book•
08 Jun 2010
TL;DR: The idea of S-FEM is to create a finite-dimensional space for the creation of shape functions as discussed by the authors, which can be used to solve problems in engineering such as failure, uniqueness, error, and convergence.
Abstract: Introduction Physical Problems in Engineering Numerical Techniques: Practical Solution Tools Why S-FEM? The Idea of S-FEM Key Techniques Used in S-FEM S-FEM Models and Properties Some Historical Notes Outline of the Book Basic Equations for Solid Mechanics Equilibrium Equation: In Stresses Constitutive Equation Compatibility Equation Equilibrium Equation: In Displacements Equations in Matrix Form Boundary Conditions Some Standard Default Conventions and Notations The Finite Element Method General Procedure of FEM Proper Spaces Weak Formulation and Properties of the Solution Domain Discretization: Creation of Finite-Dimensional Space Creation of Shape Functions Displacement Function Creation Strain Evaluation Formulation of the Discretized System of Equations FEM Solution: Existence, Uniqueness, Error, and Convergence Some Other Properties of the FEM Solution Linear Triangular Element (T3) Four-Node Quadrilateral Element (Q4) Four-Node Tetrahedral Element (T4) Eight-Node Hexahedral Element (H8) Gauss Integration Fundamental Theories for S-FEM General Procedure for S-FEM Models Domain Discretization with Polygonal Elements Creating a Displacement Field: Shape Function Construction Evaluation of the Compatible Strain Field Modify/Construct the Strain Field Minimum Number of Smoothing Domains: Essential to Stability Smoothed Galerkin Weak Form Discretized Linear Algebraic System of Equations Solve the Algebraic System of Equations Error Assessment in S-FEM and FEM Models Implementation Procedure for S-FEM Models General Properties of S-FEM Models Cell-Based Smoothed FEM Cell-Based Smoothing Domain Discretized System of Equations Shape Function Evaluation Some Properties of CS-FEM Stability of CS-FEM and nCS-FEM Standard Patch Test: Accuracy Selective CS-FEM: Volumetric Locking Free Numerical Examples Node-Based Smoothed FEM Introduction Creation of Node-Based Smoothing Domains Formulation of NS-FEM Evaluation of Shape Function Values Properties of NS-FEM An Adaptive NS-FEM Using Triangular Elements Numerical Examples Edge-Based Smoothed FEM Introduction Creation of Edge-Based Smoothing Domains Formulation of the ES-FEM Evaluation of the Shape Function Values in the ES-FEM A Smoothing-Domain-Based Selective ES/NS-FEM Properties of the ES-FEM Numerical Examples Face-Based Smoothed FEM Introduction Face-Based Smoothing Domain Creation Formulation of FS-FEM-T4 A Smoothing-Domain-Based Selective FS/NS-FEM-T4 Model Stability, Accuracy, and Mesh Sensitivity Numerical Examples The alphaFEM Introduction Idea of alphaFEM-T3 and alphaFEM-T4 alphaFEM-T3 and alphaFEM-T4 for Nonlinear Problems Implementation and Patch Tests Numerical Examples S-FEM for Fracture Mechanics Introduction Singular Stress Field Creation at the Crack-Tip Possible sS-FEM Methods sNS-FEM Models sES-FEM Models Stiffness Matrix Evaluation J-Integral and SIF Evaluation Interaction Integral Method for Mixed Mode Numerical Examples Solved Using sES-FEM-T3 Numerical Examples Solved Using sNS-FEM-T3 S-FEM for Viscoelastoplasticity Introduction Strong Formulation for Viscoelastoplasticity FEM for Viscoelastoplasticity: A Dual Formulation S-FEM for Viscoelastoplasticity: A Dual Formulation A Posteriori Error Estimation Numerical Examples ES-FEM for Plates Introduction Weak Form for the Reissner-Mindlin Plate FEM Formulation for the Reissner-Mindlin Plate ES-FEM-DSG3 for the Reissner-Mindlin Plate Numerical Examples: Patch Test Numerical Examples: Static Analysis Numerical Examples: Free Vibration of Plates Numerical Examples: Buckling of Plates S-FEM for Piezoelectric Structures Introduction Galerkin Weak Form for Piezoelectrics Finite Element Formulation for the Piezoelectric Problem S-FEM for the Piezoelectric Problem Numerical Results S-FEM for Heat Transfer Problems Introduction Strong-Form Equations for Heat Transfer Problems Boundary Conditions Weak Forms for Heat Transfer Problems FEM Equations S-FEM Equations Evaluation of the Smoothed Gradient Matrix Numerical Example Bioheat Transfer Problems S-FEM for Acoustics Problems Introduction Mathematical Model of Acoustics Problems Weak Forms for Acoustics Problems FEM Equations S-FEM Equations Error in a Numerical Model Numerical Examples Index References appear at the end of each chapter.
352 citations
TL;DR: This paper revisits a powerful discretization technique, the Proper Generalized Decomposition—PGD, illustrating its ability for solving highly multidimensional models.
Abstract: This paper revisits a powerful discretization technique, the Proper Generalized Decomposition—PGD, illustrating its ability for solving highly multidimensional models. This technique operates by constructing a separated representation of the solution, such that the solution complexity scales linearly with the dimension of the space in which the model is defined, instead the exponentially-growing complexity characteristic of mesh based discretization strategies. The PGD makes possible the efficient solution of models defined in multidimensional spaces, as the ones encountered in quantum chemistry, kinetic theory description of complex fluids, genetics (chemical master equation), financial mathematics, … but also those, classically defined in the standard space and time, to which we can add new extra-coordinates (parametric models, …) opening numerous possibilities (optimization, inverse identification, real time simulations, …).
351 citations
TL;DR: A general time-discrete framework to design asymptotic-preserving schemes for initial value problem of the Boltzmann kinetic and related equations, which can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved.
340 citations
TL;DR: A new discretization scheme for Maxwell equations in two space dimension based on the use of bivariate B-splines spaces suitably adapted to electromagnetics and provides regular discrete solutions of Maxwell equations.
TL;DR: A physically consistent phase-field model that admits an energy law is proposed, and several energy stable, efficient, and accurate time discretization schemes for the coupled nonlinear phase- field model are constructed and analyzed.
Abstract: Modeling and numerical approximation of two-phase incompressible flows with different densities and viscosities are considered. A physically consistent phase-field model that admits an energy law is proposed, and several energy stable, efficient, and accurate time discretization schemes for the coupled nonlinear phase-field model are constructed and analyzed. Ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for problems with large density and viscosity ratios.
TL;DR: In this paper, the authors unify all Euler fixes into a single general framework and introduce a new full truncation scheme, tailored to minimize the positive bias found when pricing European options.
Abstract: Using an Euler discretization to simulate a mean-reverting CEV process gives rise to the problem that while the process itself is guaranteed to be nonnegative, the discretization is not. Although an exact and efficient simulation algorithm exists for this process, at present this is not the case for the CEV-SV stochastic volatility model, with the Heston model as a special case, where the variance is modelled as a mean-reverting CEV process. Consequently, when using an Euler discretization, one must carefully think about how to fix negative variances. Our contribution is threefold. Firstly, we unify all Euler fixes into a single general framework. Secondly, we introduce the new full truncation scheme, tailored to minimize the positive bias found when pricing European options. Thirdly and finally, we numerically compare all Euler fixes to recent quasi-second order schemes of Kahl and Jackel, and Ninomiya and Victoir, as well as to the exact scheme of Broadie and Kaya. The choice of fix is found to be extre...
TL;DR: In this paper, a new scalar hyperbolic partial differential equation (PDE) model for traffic velocity evolution on highways, based on the seminal Lighthill-Whitham-Richards (LWR) PDE for density, is presented.
Abstract: This article is motivated by the practical problem of highway traffic estimation using velocity measurements from GPS enabled mobile devices such as cell phones. In order to simplify the estimation procedure, a velocity model for highway traffic is constructed, which results in a dynamical system in which the observation operator is linear. This article presents a new scalar hyperbolic partial differential equation (PDE) model for traffic velocity evolution on highways, based on the seminal Lighthill-Whitham-Richards (LWR) PDE for density. Equivalence of the solution of the new velocity PDE and the solution of the LWR PDE is shown for quadratic flux functions. Because this equivalence does not hold for general flux functions, a discretized model of velocity evolution based on the Godunov scheme applied to the LWR PDE is proposed. Using an explicit instantiation of the weak boundary conditions of the PDE, the discrete velocity evolution model is generalized to a network, thus making the model applicable to arbitrary highway networks. The resulting velocity model is a nonlinear and nondifferentiable discrete time dynamical system with a linear observation operator, for which a Monte Carlo based ensemble Kalman filtering data
TL;DR: In this paper, the authors introduce a new technique for the error analysis of hybridizable discontinuous Galerkin (HDG) methods, which relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods.
Abstract: We introduce a new technique for the error analysis of hybridizable discontinuous Galerkin (HDG) methods. The technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods. This renders the analysis of the projections of the discretization errors simple and concise. By showing that these projections of the errors are bounded in terms of the distance between the solution and its projection, our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection. We illustrate the technique on a specific HDG method applied to a model second-order elliptic problem.
TL;DR: This study makes the first attempt to apply the Kansa method in the solution of the time fractional diffusion equations, in which the MultiQuadrics and thin plate spline serve as the radial basis function.
Abstract: This study makes the first attempt to apply the Kansa method in the solution of the time fractional diffusion equations, in which the MultiQuadrics and thin plate spline serve as the radial basis function. In the discretization formulation, the finite difference scheme and the Kansa method are respectively used to discretize time fractional derivative and spatial derivative terms. The numerical solutions of one- and two-dimensional cases are presented and discussed, which agree well with the corresponding analytical solution.
TL;DR: A hybridizable discontinuous Galerkin method for Stokes flow that reduces the globally coupled unknowns to the numerical trace of the velocity and the mean of the pressure on the element boundaries, thereby leading to a significant reduction in the size of the resulting matrix.
TL;DR: A new FFT-based scheme is proposed which is as simple as the basic scheme, while remaining valid for infinite contrasts, and provides an energetically consistent rule for the homogenization of boundary voxels.
TL;DR: In this paper, the authors investigated the well-posedness of a coupled Stokes-Darcy model with Beavers-Joseph interface boundary conditions, and established the wellposedness via an appropriate time discretization of the problem and a novel scaling of the system under an isotropic media assumption.
Abstract: We investigate the well-posedness of a coupled Stokes-Darcy model with Beavers-Joseph interface boundary conditions. In the steady-state case, the well-posedness is established under the assumption of a small coefficient in the Beavers-Joseph interface boundary condition. In the time-dependent case, the well-posedness is established via an appropriate time discretization of the problem and a novel scaling of the system under an isotropic media assumption. Such coupled systems are crucial to the study of subsurface flow problems, in particular, flows in karst aquifers.
TL;DR: Computational experiments confirm robustness of the algorithm with respect to its internal parameters and demonstrate significant increase of the convergence rate for problems with high-contrast coefficients at a low overhead per iteration.
TL;DR: In this article, a bilinear W 2 formulation for solid mechanics problems is proposed, which is based on the G space theory and is shown to be spatially stable and convergent to exact solutions.
Abstract: In part I of this paper, we have established the G space theory and fundamentals for W 2 formulation. Part II focuses on the applications of the G space theory to formulate W 2 models for solid mechanics problems. We first define a bilinear form, prove some of the important properties, and prove that the W 2 formulation will be spatially stable, and convergent to exact solutions. We then present examples of some of the possible W 2 models including the SFEM, NS-FEM, ES-FEM, NS-PIM, ES-PIM, and CS-PIM. We show the major properties of these models: (1) they are variationally consistent in a conventional sense, if the solution is sought in a proper H space (compatible cases); (2) They pass the standard patch test when the solution is sought in a proper G space with discontinuous functions (incompatible cases); (3) the stiffness of the discretized model is reduced compared with the finite element method (FEM) model and possibly to the exact model, allowing us to obtain upper bound solutions with respect to both the FEM and the exact solutions and (4) the W 2 models are less sensitive to the quality of the mesh, and triangular meshes can be used without any accuracy problems. These properties and theories have been confirmed numerically via examples solved using a number of W 2 models including compatible and incompatible cases. We shall see that the G space theory and the W 2 forms can formulate a variety of stable and convergent numerical methods with the FEM as one special case.
TL;DR: In this article, the authors report on some recent results for mean field models in discrete time with a finite number of states, and address existence, uniqueness and exponential convergence, and check to equilibrium results.
TL;DR: A spectral boundary integral method for simulating large numbers of blood cells flowing in complex geometries is developed and demonstrated and is shown to reproduce the well-known non-monotonic dependence of the effective viscosity on the tube diameter.
TL;DR: This paper presents a multiresolution topology optimization (MTOP) scheme to obtain high resolution designs with relatively low computational cost and demonstrates via various two- and three-dimensional numerical examples that the resolution of the design can be significantly improved without refining the finite element mesh.
Abstract: 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.
TL;DR: In this article, the authors considered a model class of second order, linear, parametric, elliptic PDEs in a bounded domain D with coecients depending on possibly countably many parameters and showed that the dependence of the solution on the parameters in the diusion coecient is analytically smooth.
Abstract: Parametric partial dierential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo when solving stochastic elliptic problems. This paper considers a model class of second order, linear, parametric, elliptic PDEs in a bounded domain D with coecients depending on possibly countably many parameters. It shows that the dependence of the solution on the parameters in the diusion coecient is analytically smooth. This analyticity is then exploited to prove that under very weak assumptions on the diusion coecients, the entire family of solutions to such equations can be simultaneously approximated by multivariate polynomials (in the parameters) with coecients taking values in the Hilbert space V = H 1 0(D) of weak solutions of the elliptic problem with a controlled number of terms N. The convergence rate in terms of N does not depend on the number of parameters in V which may be countable, therefore breaking the curse of dimensionality. The discretization of the coecients from a family of continuous, piecewise linear Finite Element functions in D is shown to yield finite dimensional approximations whose convergence rate in terms of the overall number Ndof of degrees of freedom is the minimum of the convergence rates aorded by the best N-term sequence approximations in the parameter space and the rate of Finite Element approximations in D for a single instance of the parametric problem.
TL;DR: This paper shows how to achieve a full and strong coupling between anisotropic mesh adaptation and goal-oriented error estimate in three steps based on a careful analysis of the contributions of the implicit error and of the interpolation error.
TL;DR: The numerical results indicate that this reconstruction-based discontinuous Galerkin (RDG) method is able to deliver the same accuracy as the well-known Bassi-Rebay II scheme, at a half of its computing costs for the discretization of the viscous fluxes in the Navier-Stokes equations, clearly demonstrating its superior performance over the existing DG methods.
TL;DR: This paper presents weak second and third order schemes for the Cox-Ingersoll-Ross process, without any restriction on its parameters, and gives a general recursive construction method to get weak second-order schemes that extends the one introduced by Ninomiya and Victoir~\cite{NV.
Abstract: This paper presents weak second and third order schemes for the Cox-Ingersoll-Ross (CIR) process, without any restriction on its parameters. At the same time, it gives a general recursive construction method to get weak second-order schemes that extends the one introduced by Ninomiya and Victoir~\cite{NV}. Combining these both results, this allows to propose a second-order scheme for more general affine diffusions. Simulation examples are given to illustrate the convergence of these schemes on CIR and Heston models. Algorithms are stated in a pseudocode language.
TL;DR: A new computational tool for computing equilibria based on an L^2 relaxation flow for the total energy in which the line energy is approximated by a surface Ginzburg-Landau phase field functional is proposed.
TL;DR: In this article, a unified finite element approach to fully coupled cardiac electromechanics is proposed, where the intrinsic coupling arises from both the excitation-induced contraction of cardiac cells and the deformation-induced generation of current due to the opening of ion channels.
Abstract: This manuscript is concerned with a novel, unified finite element approach to fully coupled cardiac electromechanics The intrinsic coupling arises from both the excitation-induced contraction of cardiac cells and the deformation-induced generation of current due to the opening of ion channels In contrast to the existing numerical approaches suggested in the literature, which devise staggered algorithms through distinct numerical methods for the respective electrical and mechanical problems, we propose a fully implicit, entirely finite element-based modular approach To this end, the governing differential equations that are coupled through constitutive equations are recast into the corresponding weak forms through the conventional isoparametric Galerkin method The resultant non-linear weighted residual terms are then consistently linearized The system of coupled algebraic equations obtained through discretization is solved monolithically The put-forward modular algorithmic setting leads to an unconditionally stable and geometrically flexible framework that lays a firm foundation for the extension of constitutive equations towards more complex ionic models of cardiac electrophysiology and the strain energy functions of cardiac mechanics The performance of the proposed approach is demonstrated through three-dimensional illustrative initial boundary-value problems that include a coupled electromechanical analysis of a biventricular generic heart model