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Showing papers on "Discretization published in 2012"


Journal ArticleDOI
TL;DR: In this paper, the authors presented an infinitesimal-strain version of a formulation based on fast Fourier transforms (FFT) for the prediction of micromechanical fields in polycrystals deforming in the elasto-viscoplastic (EVP) regime.

441 citations


Journal ArticleDOI
TL;DR: This work addresses the solution of large-scale statistical inverse problems in the framework of Bayesian inference with a so-called Stochastic Monte Carlo method.
Abstract: We address the solution of large-scale statistical inverse problems in the framework of Bayesian inference. The Markov chain Monte Carlo (MCMC) method is the most popular approach for sampling the posterior probability distribution that describes the solution of the statistical inverse problem. MCMC methods face two central difficulties when applied to large-scale inverse problems: first, the forward models (typically in the form of partial differential equations) that map uncertain parameters to observable quantities make the evaluation of the probability density at any point in parameter space very expensive; and second, the high-dimensional parameter spaces that arise upon discretization of infinite-dimensional parameter fields make the exploration of the probability density function prohibitive. The challenge for MCMC methods is to construct proposal functions that simultaneously provide a good approximation of the target density while being inexpensive to manipulate. Here we present a so-called Stoch...

411 citations


Journal ArticleDOI
TL;DR: The k-space pseudospectral method is used to reduce the number of grid points required per wavelength for accurate simulations of nonlinear ultrasound propagation through tissue realistic media, and increases the accuracy of the gradient calculation and relaxes the requirement for dense computational grids compared to conventional finite difference methods.
Abstract: The simulation of nonlinear ultrasound propagation through tissue realistic media has a wide range of practical applications. However, this is a computationally difficult problem due to the large size of the computational domain compared to the acoustic wavelength. Here, the k-space pseudospectral method is used to reduce the number of grid points required per wavelength for accurate simulations. The model is based on coupled first-order acoustic equations valid for nonlinear wave propagation in heterogeneous media with power law absorption. These are derived from the equations of fluid mechanics and include a pressure-density relation that incorporates the effects of nonlinearity, power law absorption, and medium heterogeneities. The additional terms accounting for convective nonlinearity and power law absorption are expressed as spatial gradients making them efficient to numerically encode. The governing equations are then discretized using a k-space pseudospectral technique in which the spatial gradients are computed using the Fourier-collocation method. This increases the accuracy of the gradient calculation and thus relaxes the requirement for dense computational grids compared to conventional finite difference methods. The accuracy and utility of the developed model is demonstrated via several numerical experiments, including the 3D simulation of the beam pattern from a clinical ultrasound probe.

407 citations


Journal ArticleDOI
TL;DR: The adaptive mesh refinement (AMR) as mentioned in this paper implementation of the PLUTO code for solving the equations of classical and relativistic magnetohydrodynamics (MHD and RMHD) exploits, in addition to the static grid version of the code, the distributed infrastructure of the CHOMBO library for multidimensional parallel computations over block-structured, adaptively refined grids.
Abstract: We present a description of the adaptive mesh refinement (AMR) implementation of the PLUTO code for solving the equations of classical and special relativistic magnetohydrodynamics (MHD and RMHD). The current release exploits, in addition to the static grid version of the code, the distributed infrastructure of the CHOMBO library for multidimensional parallel computations over block-structured, adaptively refined grids. We employ a conservative finite-volume approach where primary flow quantities are discretized at the cell center in a dimensionally unsplit fashion using the Corner Transport Upwind method. Time stepping relies on a characteristic tracing step where piecewise parabolic method, weighted essentially non-oscillatory, or slope-limited linear interpolation schemes can be handily adopted. A characteristic decomposition-free version of the scheme is also illustrated. The solenoidal condition of the magnetic field is enforced by augmenting the equations with a generalized Lagrange multiplier providing propagation and damping of divergence errors through a mixed hyperbolic/parabolic explicit cleaning step. Among the novel features, we describe an extension of the scheme to include non-ideal dissipative processes, such as viscosity, resistivity, and anisotropic thermal conduction without operator splitting. Finally, we illustrate an efficient treatment of point-local, potentially stiff source terms over hierarchical nested grids by taking advantage of the adaptivity in time. Several multidimensional benchmarks and applications to problems of astrophysical relevance assess the potentiality of the AMR version of PLUTO in resolving flow features separated by large spatial and temporal disparities.

398 citations


Journal ArticleDOI
TL;DR: The approach represents beliefs (the distributions of the robot’s state estimate) by Gaussian distributions and is applicable to robot systems with non-linear dynamics and observation models and in simulation for holonomic and non-holonomic robots maneuvering through environments with obstacles with noisy and partial sensing.
Abstract: We present a new approach to motion planning under sensing and motion uncertainty by computing a locally optimal solution to a continuous partially observable Markov decision process (POMDP). Our approach represents beliefs (the distributions of the robot's state estimate) by Gaussian distributions and is applicable to robot systems with non-linear dynamics and observation models. The method follows the general POMDP solution framework in which we approximate the belief dynamics using an extended Kalman filter and represent the value function by a quadratic function that is valid in the vicinity of a nominal trajectory through belief space. Using a belief space variant of iterative LQG (iLQG), our approach iterates with second-order convergence towards a linear control policy over the belief space that is locally optimal with respect to a user-defined cost function. Unlike previous work, our approach does not assume maximum-likelihood observations, does not assume fixed estimator or control gains, takes into account obstacles in the environment, and does not require discretization of the state and action spaces. The running time of the algorithm is polynomial (O[n6]) in the dimension n of the state space. We demonstrate the potential of our approach in simulation for holonomic and non-holonomic robots maneuvering through environments with obstacles with noisy and partial sensing and with non-linear dynamics and observation models.

325 citations


03 Oct 2012
TL;DR: A variational free-discontinuity formulation of brittle fracture was given by Francfort and Marigo as discussed by the authors, where the total energy is minimized with respect to the crackgeometry and the displacement field simultaneously.
Abstract: A variational free-discontinuity formulation of brittle fracture was given by Francfortand Marigo [1], where the total energy is minimized with respect to the crackgeometry and the displacement field simultaneously. The entire evolution of cracksincluding their initiation and branching is determined by this minimization principlerequiring no further criterion. However, a direct numerical discretization of themodel faces considerable difficulties as the displacement field is discontinuous inthe presence of cracks.

313 citations


Journal ArticleDOI
TL;DR: In this article, the authors developed a framework for fluid-structure interaction (FSI) modeling and simulation with emphasis on isogeometric analysis (IGA) and non-matching fluid -structure interface discretizations.

313 citations


Journal ArticleDOI
TL;DR: This work gives a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly.

268 citations


Journal ArticleDOI
TL;DR: This work considers a control volume discretization with a multi-point flux approximation to model Discrete Fracture-Matrix systems for anisotropic and fractured porous media in two and three spatial dimensions and explicitly account for the fractures by representing them as hybrid cells between the matrix cells.

254 citations


Book
26 Mar 2012
TL;DR: In this paper, the authors present mathematical strategies for filtering turbulent signal with model error, including the Kalman filter for vector systems, reduced filters and a three-dimensional toy model.
Abstract: Preface 1. Introduction and overview: mathematical strategies for filtering turbulent systems Part I. Fundamentals: 2. Filtering a stochastic complex scalar: the prototype test problem 3. The Kalman filter for vector systems: reduced filters and a three-dimensional toy model 4. Continuous and discrete Fourier series and numerical discretization Part II. Mathematical Guidelines for Filtering Turbulent Signals: 5. Stochastic models for turbulence 6. Filtering turbulent signals: plentiful observations 7. Filtering turbulent signals: regularly spaced sparse observations 8. Filtering linear stochastic PDE models with instability and model error Part III. Filtering Turbulent Nonlinear Dynamical Systems: 9. Strategies for filtering nonlinear systems 10. Filtering prototype nonlinear slow-fast systems 11. Filtering turbulent nonlinear dynamical systems by finite ensemble methods 12. Filtering turbulent nonlinear dynamical systems by linear stochastic models 13. Stochastic parameterized extended Kalman filter for filtering turbulent signal with model error 14. Filtering turbulent tracers from partial observations: an exactly solvable test model 15. The search for efficient skilful particle filters for high dimensional turbulent dynamical systems References Index.

236 citations


Journal ArticleDOI
TL;DR: This work shows that the flexibility of the discontinuous Galerkin (dG) discretization can be fruitfully exploited to implement numerical solution strategies based on the use of elements with very general shapes, and proposes a new h-adaptive technique based on agglomeration coarsening of a fine mesh.

Journal ArticleDOI
TL;DR: A thermodynamically consistent four-species model of tumor growth on the basis of the continuum theory of mixtures, unique to this model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation.
Abstract: In this paper, we develop a thermodynamically consistent four-species model of tumor growth on the basis of the continuum theory of mixtures. Unique to this model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models. A mixed finite element spatial discretization is developed and implemented to provide numerical results demonstrating the range of solutions this model can produce. A time-stepping algorithm is then presented for this system, which is shown to be first order accurate and energy gradient stable. The results of an array of numerical experiments are presented, which demonstrate a wide range of solutions produced by various choices of model parameters.

Journal ArticleDOI
TL;DR: The fractional diffusion equation is discretized by the implicit finite difference scheme with the shifted Grunwald formula and the coefficient matrix possesses the Toeplitz-like structure and a multigrid method is proposed to solve the resulting system.

Journal ArticleDOI
TL;DR: A fast and yet accurate solution method for the implicit finite difference discretization of space-fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient matrices is developed.
Abstract: Fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Because of the nonlocal property of fractional differential operators, numerical methods for space-fractional diffusion equations generate complicated dense or full coefficient matrices. Consequently, these numerical methods were traditionally solved by Gaussian elimination, which requires computational work of $O(N^3)$ per time step and $O(N^2)$ of memory, where $N$ is the number of spatial grid points in the discretization. The significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two- and especially three-dimensional space-fractional diffusion equations. We develop a fast and yet accurate solution method for the implicit finite difference discretization of space-fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient ma...

Journal ArticleDOI
TL;DR: This paper presents a general framework for high-order Lagrangian discretization of these compressible shock hydrodynamics equations using curvilinear finite elements for any finite dimensional approximation of the kinematic and thermodynamic fields.
Abstract: The numerical approximation of the Euler equations of gas dynamics in a movingLagrangian frame is at the heart of many multiphysics simulation algorithms. In this paper, we present a general framework for high-order Lagrangian discretization of these compressible shock hydrodynamics equations using curvilinear finite elements. This method is an extension of the approach outlined in [Dobrev et al., Internat. J. Numer. Methods Fluids, 65 (2010), pp. 1295--1310] and can be formulated for any finite dimensional approximation of the kinematic and thermodynamic fields, including generic finite elements on two- and three-dimensional meshes with triangular, quadrilateral, tetrahedral, or hexahedral zones. We discretize the kinematic variables of position and velocity using a continuous high-order basis function expansion of arbitrary polynomial degree which is obtained via a corresponding high-order parametric mapping from a standard reference element. This enables the use of curvilinear zone geometry, higher-ord...

Journal ArticleDOI
TL;DR: Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, an approximation of the original matrix is embedded into a larger but highly structured sparse one that allows fast factorization and application of the inverse.
Abstract: We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the original matrix into a larger but highly structured sparse one that allows fast factorization and application of the inverse. The algorithm extends the Martinsson--Rokhlin method developed for 2D boundary integral equations and proceeds in two phases: a precomputation phase, consisting of matrix compression and factorization, followed by a solution phase to apply the matrix inverse. For boundary integral equations which are not too oscillatory, e.g., based on the Green functions for the Laplace or low-frequency Helmholtz equations, both phases typically have complexity $\mathcal{O} (N)$ in two dimensions, where $N$ is the number of discretization points. In our current implementation, the corresponding costs in three dimensions are $\mathcal{O} (N^{3/2})$ and $\mathcal{O} (...

Journal ArticleDOI
TL;DR: In-plane–out-of-plane separated representation of the involved fields within the context of the Proper Generalized Decomposition allows solving the fully 3D model by keeping a 2D characteristic computational complexity, without affecting the solvability of the resulting multidimensional model.

Journal ArticleDOI
TL;DR: In this paper, a three-dimensional mortar-based frictional contact treatment in isogeometric analysis with NURBS is presented in the finite deformation regime, where the contact integrals are evaluated through a mortar approach where the geometrical and frictional contacts constraints are treated through a projection to control point quantities.

Journal ArticleDOI
TL;DR: Two new gradient schemes are constructed which have the advantage of a small stencil in discretization schemes for diffusing flows in heterogeneous anisotropic porous media.
Abstract: In this paper, we study some discretization schemes for diffusing flows in heterogeneous anisotropic porous media. We first introduce the notion of gradient scheme, and show that several existing schemes fall in this framework. Then, we construct two new gradient schemes which have the advantage of a small stencil. Numerical results obtained for real reservoir meshes, show the efficiency of the new schemes, compared to existing ones.

Journal ArticleDOI
TL;DR: This work presents a novel discretization scheme that adaptively and systematically builds the rapid oscillations of the Kohn-Sham orbitals around the nuclei as well as environmental effects into the basis functions.

Journal ArticleDOI
TL;DR: Finite difference methods for the Gross-Pitaevskii equation with an angular momentum rotation term in two and three dimensions are analyzed and error bounds on the errors between the mass and energy in the discretized level and their corresponding continuous counterparts are derived.
Abstract: We analyze finite difference methods for the Gross-Pitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate, for the conservative Crank-Nicolson finite difference (CNFD) method and semi-implicit finite difference (SIFD) method, at the order of O(h2 + τ2) in the l2-norm and discrete H1-norm with time step τ and mesh size h. Besides the standard techniques of the energy method, the key technique in the analysis for the SIFD method is to use the mathematical induction, and resp., for the CNFD method is to obtain a priori bound of the numerical solution in the l∞-norm by using the inverse inequality and the l2-norm error estimate. In addition, for the SIFD method, we also derive error bounds on the errors between the mass and energy in the discretized level and their corresponding continuous counterparts, respectively, which are at the same order of the convergence rate as that of the numerical solution itself. Finally, numerical results are reported to confirm our error estimates of the numerical methods.

Journal ArticleDOI
TL;DR: In this article, NURBS-based isogeometric analysis is applied to 3D frictionless large deformation contact problems, where the contact constraints are treated with a mortar-based approach combined with a simplified integration method avoiding segmentation of the contact surfaces, and the discretization of the continuum is performed with arbitrary order nurBS and Lagrange polynomial elements.
Abstract: NURBS-based isogeometric analysis is applied to 3D frictionless large deformation contact problems. The contact constraints are treated with a mortar-based approach combined with a simplified integration method avoiding segmentation of the contact surfaces, and the discretization of the continuum is performed with arbitrary order NURBS and Lagrange polynomial elements. The contact constraints are satisfied exactly with the augmented Lagrangian formulation proposed by Alart and Curnier, whereby a Newton-like solution scheme is applied to solve the saddle point problem simultaneously for displacements and Lagrange multipliers. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. In both small and large deformation cases, the quality of the contact pressures is shown to improve significantly over that achieved with Lagrange discretizations. In large deformation and large sliding examples, the NURBS discretization provides an improved smoothness of the traction history curves. In both cases, increasing the order of the discretization is either beneficial or not influential when using isogeometric analysis, whereas it affects results negatively for Lagrange discretizations.

Journal ArticleDOI
TL;DR: A difference scheme combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping is proposed and analyzed and it is indicated that the compact ADI scheme reduces the storage requirement and CPU time successfully.
Abstract: In this paper, we consider the numerical method for solving the two-dimensional fractional diffusion-wave equation with a time fractional derivative of order $\alpha$ ($1<\alpha<2$). A difference scheme combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping is proposed and analyzed. The unconditional stability and $H^1$ norm convergence of the scheme are proved rigorously. The convergence order is $\mathcal {O}(\tau^{3-\alpha}+h_1^4+h^4_2),$ where $\tau$ is the temporal grid size and $h_1,h_2$ are spatial grid sizes in the $x$ and $y$ directions, respectively. In addition, a Crank--Nicolson ADI scheme is presented and the corresponding error estimates are also established. Numerical results are presented to support our theoretical analysis and indicate that the compact ADI scheme reduces the storage requirement and CPU time successfully.

Journal ArticleDOI
TL;DR: A model that uses local discretization on a circle around virtual pedestrians that allows for movement in arbitrary directions, only limited by the chosen optimization algorithm and numerical resolution is developed.
Abstract: Is there a way to describe pedestrian movement with simple rules, as in a cellular automaton, but without being restricted to a cellular grid? Inspired by the natural stepwise movement of humans, we develop a model that uses local discretization on a circle around virtual pedestrians. This allows for movement in arbitrary directions, only limited by the chosen optimization algorithm and numerical resolution. The radii of the circles correspond to the step lengths of pedestrians and thus are model parameters, which must be derived from empirical observation. Therefore, we conducted a controlled experiment, collected empirical data for step lengths in relation with different speeds, and used the findings in our model. We complement the model with a simple calibration algorithm that allows reproducing known density-velocity relations, which constitutes a proof of concept. Further validation of the model is achieved by reenacting an evacuation scenario from experimental research. The simulated egress times match the values reported for the experiment very well. A new normalized measure for space occupancy serves to visualize the results.

Journal ArticleDOI
TL;DR: One and two-dimensional numerical results provide a validation of the Asymptotic-Preserving 'all-speed' properties.

Journal ArticleDOI
TL;DR: This paper presents a kinodynamic motion planner, i.e., Kinodynamic Motion Planning by Interior-Exterior Cell Exploration (KPIECE), which is specifically designed for systems with complex dynamics, where integration backward in time is not possible, and speed of computation is important.
Abstract: This paper presents a kinodynamic motion planner, i.e., Kinodynamic Motion Planning by Interior-Exterior Cell Exploration (KPIECE), which is specifically designed for systems with complex dynamics, where integration backward in time is not possible, and speed of computation is important. A grid-based discretization is used to estimate the coverage of the state space. The coverage estimates help the planner detect the less-explored areas of the state space. An important characteristic of this discretization is that it keeps track of the boundary of the explored region of the state space and focuses exploration on the less covered parts of this boundary. Extensive experiments show that KPIECE provides significant computational gain over existing state-of-the-art methods and allows us to solve some harder, previously unsolvable problems. For some problems, KPIECE is shown to be up to two orders of magnitude faster than existing methods and use up to 40 times less memory. A shared memory parallel implementation is presented as well. This implementation provides better speedup than an embarrassingly parallel implementation by taking advantage of the evolving multicore technology.

Journal ArticleDOI
TL;DR: In this article, a robust and efficient computational method for reconstructing the elastodynamic structural response of truss, beam, and frame structures, using measured surface-strain data, is presented.

Journal ArticleDOI
01 Jul 2012
TL;DR: The thrust network method of analysis is used and an iterative nonlinear optimization algorithm is presented for efficiently approximating freeform shapes by self-supporting ones: steel/glass constructions with low moments in nodes.
Abstract: Self-supporting masonry is one of the most ancient and elegant techniques for building curved shapes. Because of the very geometric nature of their failure, analyzing and modeling such strutures is more a geometry processing problem than one of classical continuum mechanics. This paper uses the thrust network method of analysis and presents an iterative nonlinear optimization algorithm for efficiently approximating freeform shapes by self-supporting ones. The rich geometry of thrust networks leads us to close connections between diverse topics in discrete differential geometry, such as a finite-element discretization of the Airy stress potential, perfect graph Laplacians, and computing admissible loads via curvatures of polyhedral surfaces. This geometric viewpoint allows us, in particular, to remesh self-supporting shapes by self-supporting quad meshes with planar faces, and leads to another application of the theory: steel/glass constructions with low moments in nodes.

Journal ArticleDOI
TL;DR: In this paper, the generalized density evolution equation (GDEE) is derived for nonlinear stochastic systems, which is a unified basis for the probability density evolution equations holding for different types of systems.

Journal ArticleDOI
TL;DR: In this article, the authors studied the unconditional convergence and error estimates of a Galerkin-mixed FEM with the linearized semi-implicit Euler time-discrete scheme for the equations of incompressible miscible flow in porous media.
Abstract: In this paper, we study the unconditional convergence and error estimates of a Galerkin-mixed FEM with the linearized semi-implicit Euler time-discrete scheme for the equations of incompressible miscible flow in porous media. We prove that the optimal $L^2$ error estimates hold without any time-step (convergence) condition, while all previous works require certain time-step condition. Our theoretical results provide a new understanding on commonly-used linearized schemes for nonlinear parabolic equations. The proof is based on a splitting of the error function into two parts: the error from the time discretization of the PDEs and the error from the finite element discretization of corresponding time-discrete PDEs. The approach used in this paper is applicable for more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations.