Showing papers in "Archives of Computational Methods in Engineering in 2010"
TL;DR: An overview on the SPH method and its recent developments is presented, including the need for meshfree particle methods, and advantages of SPH, and several important numerical aspects.
Abstract: Smoothed particle hydrodynamics (SPH) is a meshfree particle method based on Lagrangian formulation, and has been widely applied to different areas in engineering and science. This paper presents an overview on the SPH method and its recent developments, including (1) the need for meshfree particle methods, and advantages of SPH, (2) approximation schemes of the conventional SPH method and numerical techniques for deriving SPH formulations for partial differential equations such as the Navier-Stokes (N-S) equations, (3) the role of the smoothing kernel functions and a general approach to construct smoothing kernel functions, (4) kernel and particle consistency for the SPH method, and approaches for restoring particle consistency, (5) several important numerical aspects, and (6) some recent applications of SPH. The paper ends with some concluding remarks.
1,398 citations
TL;DR: A review of methods applicable to the study of masonry historical construction, encompassing both classical and advanced ones, is presented in this paper, where the main available strategies, including limit analysis, simplified methods, FEM macro- or micro-modeling and discrete element methods (DEM), are considered with regard to their realism, computer efficiency, data availability and real applicability to large structures.
Abstract: A review of methods applicable to the study of masonry historical construction, encompassing both classical and advanced ones, is presented. Firstly, the paper offers a discussion on the main challenges posed by historical structures and the desirable conditions that approaches oriented to the modeling and analysis of this type of structures should accomplish. Secondly, the main available methods which are actually used for study masonry historical structures are referred to and discussed. The main available strategies, including limit analysis, simplified methods, FEM macro- or micro-modeling and discrete element methods (DEM) are considered with regard to their realism, computer efficiency, data availability and real applicability to large structures. A set of final considerations are offered on the real possibility of carrying out realistic analysis of complex historic masonry structures. In spite of the modern developments, the study of historical buildings is still facing significant difficulties linked to computational effort, possibility of input data acquisition and limited realism of methods.
504 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: The design and implementation of a framework for building multi-disciplinary finite element programs, which contains several tools for the easy implementation of finite element applications and also provides a common platform for the natural interaction of different applications is described.
Abstract: The objective of this work is to describe the design and implementation of a framework for building multi-disciplinary finite element programs. The main goals are generality, reusability, extendibility, good performance and memory efficiency. Another objective is preparing the code structure for team development to ensure the easy collaboration of experts in different fields in the development of multi-disciplinary applications. Kratos, the framework described in this work, contains several tools for the easy implementation of finite element applications and also provides a common platform for the natural interaction of different applications. To achieve this, an innovative variable base interface is designed and implemented. This interface is used at different levels of abstraction and showed to be very clear and extendible. A very efficient and flexible data structure and an extensible IO are created to overcome difficulties in dealing with multi-disciplinary problems. Several other concepts in existing works are also collected and adapted to coupled problems. The use of an interpreter, of different data layouts and variable number of dofs per node are examples of such approach. In order to minimize the possible conflicts arising in the development, a kernel and application approach is used. The code is structured in layers to reflect the working space of developers with different fields of expertise. Details are given on the approach chosen to increase performance and efficiency. Examples of application of Kratos to different multidisciplinary problems are presented in order to demonstrate the applicability and efficiency of the new object oriented environment.
301 citations
TL;DR: An efficient algorithm is proposed for the a priori construction of separated representations of square integrable vector-valued functions defined on a high-dimensional probability space, which are the solutions of systems of stochastic algebraic equations.
Abstract: Uncertainty quantification and propagation in physical systems appear as a critical path for the improvement of the prediction of their response. Galerkin-type spectral stochastic methods provide a general framework for the numerical simulation of physical models driven by stochastic partial differential equations. The response is searched in a tensor product space, which is the product of deterministic and stochastic approximation spaces. The computation of the approximate solution requires the solution of a very high dimensional problem, whose calculation costs are generally prohibitive. Recently, a model reduction technique, named Generalized Spectral Decomposition method, has been proposed in order to reduce these costs. This method belongs to the family of Proper Generalized Decomposition methods. It takes part of the tensor product structure of the solution function space and allows the a priori construction of a quasi optimal separated representation of the solution, which has quite the same convergence properties as a posteriori Hilbert Karhunen-Loeve decompositions. The associated algorithms only require the solution of a few deterministic problems and a few stochastic problems on deterministic reduced basis (algebraic stochastic equations), these problems being uncoupled. However, this method does not circumvent the “curse of dimensionality” which is associated with the dramatic increase in the dimension of stochastic approximation spaces, when dealing with high stochastic dimension. In this paper, we propose a marriage between the Generalized Spectral Decomposition algorithms and a separated representation methodology, which exploits the tensor product structure of stochastic functions spaces. An efficient algorithm is proposed for the a priori construction of separated representations of square integrable vector-valued functions defined on a high-dimensional probability space, which are the solutions of systems of stochastic algebraic equations.
186 citations
TL;DR: In this article, the authors report on the recent application of a now classical general reduction technique, the Reduced-Basis (RB) approach initiated by C. Prudhomme et al. in J. Fluids Eng. 124(1), 70−80, 2002, to the specific context of differential equations with random coefficients.
Abstract: We report here on the recent application of a now classical general reduction technique, the Reduced-Basis (RB) approach initiated by C. Prud’homme et al. in J. Fluids Eng. 124(1), 70–80, 2002, to the specific context of differential equations with random coefficients. After an elementary presentation of the approach, we review two contributions of the authors: in Comput. Methods Appl. Mech. Eng. 198(41–44), 3187–3206, 2009, which presents the application of the RB approach for the discretization of a simple second order elliptic equation supplied with a random boundary condition, and in Commun. Math. Sci., 2009, which uses a RB type approach to reduce the variance in the Monte-Carlo simulation of a stochastic differential equation. We conclude the review with some general comments and also discuss possible tracks for further research in the direction.
121 citations
TL;DR: This paper is a review of the developments of the Proper Generalized Decomposition (PGD) method for the resolution, using the multiscale/multiphysics LATIN method, of the nonlinear, time-dependent problems encountered in computational mechanics.
Abstract: This paper is a review of the developments of the Proper Generalized Decomposition (PGD) method for the resolution, using the multiscale/multiphysics LATIN method, of the nonlinear, time-dependent problems ((visco)plasticity, damage, …) encountered in computational mechanics. PGD leads to considerable savings in terms of computing time and storage, and makes engineering problems which would otherwise be completely out of range of industrial codes accessible.
91 citations
TL;DR: This paper proves the convergence of the Greedy Rank-One Update Algorithm and studies its relationship with the Finite Element Method for High-Dimensional Partial Differential Equations based on the tensorial product of one-dimensional bases.
Abstract: In this paper we study the convergence of the well-known Greedy Rank-One Update Algorithm. It is used to construct the rank-one series solution for full-rank linear systems. The existence of the rank one approximations is also not new, but surprisingly the focus there has been more on the applications side more that in the convergence analysis. Our main contribution is to prove the convergence of the algorithm and also we study the required rank one approximation in each step. We also give some numerical examples and describe its relationship with the Finite Element Method for High-Dimensional Partial Differential Equations based on the tensorial product of one-dimensional bases. We illustrate this situation taking as a model problem the multidimensional Poisson equation with homogeneous Dirichlet boundary condition.
91 citations
TL;DR: The PGD allows circumventing the so called curse of dimensionality that mesh based representations suffer and could allow to compute the homogenized properties for any microstructure or for any macroscopic loading history by solving a single but highly multidimensional model.
Abstract: Computational homogenization is nowadays one of the most active research topics in computational mechanics. Different strategies have been proposed, the main challenge being the computing cost induced by complex microstructures exhibiting nonlinear behaviors. Two quite tricky scenarios lie in (i) the necessity of applying the homogenization procedure for many microstructures (e.g. material microstructure evolving at the macroscopic level or stochastic microstructure); the second situation concerns the homogenization of nonlinear behaviors implying the necessity of solving microscopic problems for each macroscopic state (history independent nonlinear models) or for each macroscopic history (history dependant nonlinear models). In this paper we present some preliminary results concerning the application of Proper Generalized Decompositions—PGD—for addressing the efficient solution of homogenization problems. This numerical technique could allow to compute the homogenized properties for any microstructure or for any macroscopic loading history by solving a single but highly multidimensional model. The PGD allows circumventing the so called curse of dimensionality that mesh based representations suffer. Even if this work only describes the first steps in a very ambitious objective, many original ideas are launched that could be at the origin of impressive progresses.
61 citations
TL;DR: In this paper, an approach is developed whereby the dynamics of charged particles, accounting for their collisions, interparticle near-fields, interaction with external electromagnetic fields and coupled thermal effects are all implicitly computed in an iterative, modular, manner.
Abstract: This work addresses the modeling and simulation of charged particulate jets in the presence of electromagnetic fields. The presentation is broken into two main parts: (1) the dynamics of charged streams of particles and their interaction with electromagnetic fields and (2) the coupled thermal fields that arise within the jet. An overall model is built by assembling submodels of the various coupled physical events to form a system that is solved iteratively. Specifically, an approach is developed whereby the dynamics of charged particles, accounting for their collisions, inter-particle near-fields, interaction with external electromagnetic fields and coupled thermal effects are all computed implicitly in an iterative, modular, manner. A staggered, temporally-adaptive scheme is developed to resolve the multiple fields involved and the drastic changes in the physical configuration of the stream, for example when impacting a solid wall or strong localized electromagnetic field. Qualitative analytical results are provided to describe the effects of the electromagnetic fields and quantitative numerical results are provided for complex cases.
58 citations
TL;DR: The main goal of this work is to give a review of the Perfectly Matched Layer technique for time-harmonic problems, which involve second order elliptic equations writing in divergence form and, in particular, the Helmholtz equation at low frequency regime.
Abstract: The main goal of this work is to give a review of the Perfectly Matched Layer (PML) technique for time-harmonic problems. Precisely, we focus our attention on problems stated in unbounded domains, which involve second order elliptic equations writing in divergence form and, in particular, on the Helmholtz equation at low frequency regime. Firstly, the PML technique is introduced by means of a simple porous model in one dimension. It is emphasized that an adequate choice of the so called complex absorbing function in the PML yields to accurate numerical results. Then, in the two-dimensional case, the PML governing equation is described for second order partial differential equations by using a smooth complex change of variables. Its mathematical analysis and some particular examples are also included. Numerical drawbacks and optimal choice of the PML absorbing function are studied in detail. In fact, theoretical and numerical analysis show the advantages of using non-integrable absorbing functions. Finally, we present some relevant real life numerical simulations where the PML technique is widely and successfully used although they are not covered by the standard theoretical framework.
TL;DR: In this paper, the authors exploit the ability of the Proper Generalized Decomposition (PGD) method to handle large-dimensionality problems to transform the optimization parameters into variables of an augmented-structural problem which is solved prior to optimization.
Abstract: In structural optimization, the implicit nature of the cost function with respect to the optimization parameters, i.e. through the solution of the structural problem calculated with fixed values of these parameters, leads to prohibitive computations whatever the adopted formulation. Consequently, it yields limitations in both the number of parameters and the size of the structural problem. Moreover, some know-how is required to define a relevant structural problem and a well-behaved cost function. Here, we profit from the ability of the Proper Generalized Decomposition (PGD) method to handle large-dimensionality problems to transform the optimization parameters into variables of an augmented-structural problem which is solved prior to optimization. As a consequence, the cost function becomes explicit with respect to the parameters. As the augmented-structural problem is solved a priori, it becomes independent from the a posteriori optimization. Obviously, such approach promises numerous advantages, e.g. the solution of the structural problem can be easily analyzed to provide a guide to define the cost function and advanced optimization schemes become numerically tractable because of the easy evaluation of the cost function and its gradients.
TL;DR: In this paper, a hyperbolic convection-diffusion theory is proposed to avoid the infinite speed paradox, which predicts that disturbances can propagate at infinite speed, and a mathematical, physical and numerical analysis of the theory is presented.
Abstract: Linear parabolic diffusion theories based on Fourier’s or Fick’s laws predict that disturbances can propagate at infinite speed. Although in some applications, the infinite speed paradox may be ignored, there are many other applications in which a theory that predicts propagation at finite speed is mandatory. As a consequence, several alternatives to the linear parabolic diffusion theory, that aim at avoiding the infinite speed paradox, have been proposed over the years. This paper is devoted to the mathematical, physical and numerical analysis of a hyperbolic convection-diffusion theory.
TL;DR: Families of flux-continuous, locally conservative, finite-volume schemes are presented for solving the general geometry-permeability tensor pressure equation on structured and unstructured grids in two and three dimensions and methods for obtaining optimal discretization with minimal spurious oscillations are investigated.
Abstract: In this paper, families of flux-continuous, locally conservative, finite-volume schemes are presented for solving the general geometry-permeability tensor pressure equation on structured and unstructured grids in two and three dimensions. The schemes are applicable to the general tensor pressure equation with discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation (two-point flux) schemes when applied to full anisotropic permeability tensor flow approximation (Edwards and Rogers in Multigrids Methods, vol. 1, pp. 190–200, 1993; Edwards and Rogers in Proceedings: 4th European Conference on the Mathematics of Oil Recovery, 1994; Edwards and Rogers in Comput. Geom. 2:259–290, 1998). Full tensors arise when the local orientation of the grid is non-aligned with the principal axes of the tensor field. Full tensors may also arise when fine scale permeability distributions are upscaled to obtain gridblock-scale permeability distributions. In general full tensors arise when using any structured or unstructured grid type that departs from K-orthogonality.
TL;DR: The feasibility of the space-time separated representation method for dealing with strongly coupled multiphysics problems with different characteristic times is discussed and a new strategy to solve the nonlinear system for the basis enrichment is proposed in this paper.
Abstract: The space-time separated representation method (Ladeveze, C. R. Acad. Sci. Paris 309(II):1095–1099, 1989; Ammar et al., J. Non-Newton. Fluid Mech. 144:98–121, 2007) is here extended to solve strongly coupled multiphysics problems. The feasibility of the method for dealing with strongly coupled multiphysics problems with different characteristic times is here discussed and a new strategy to solve the nonlinear system for the basis enrichment is proposed. The method is validated in the case of a strongly coupled thermoviscoelastic model.
TL;DR: In this article, a truncated integration scheme is proposed to forecast the reduced-state variables related to the assumed separated representation, which can reduce the complexity of the integrals involved in the formulation.
Abstract: Recent developments of multidimensional solvers using separated representation make it possible to account for the multidimensionality of mechanical models in materials science when doing numerical simulations. This paper aims to extend the separated representation to inseparable equations using an efficient integration scheme. It focuses on the dependence of constitutive equations on material coefficients. Although these coefficients can be optimized using few experimental results, they are not very well known because of the natural variability of material properties. Therefore, the mechanical state can be viewed as a function depending not only on time and space variables but also on material coefficients. This is illustrated in this paper by a sensitivity analysis of the response of a sintering model with respect to variations of material coefficients. The considered variations are defined around an optimized value of coefficients adjusted by experimental results. The proposed method is an incremental method using an extension of the integration scheme developed for the Hyper Reduction method. During the incremental solution, before the adaptation of the representation, an assumed separation representation is used as a reduced-order model. We claim that a truncated integration scheme enables to forecast the reduced-state variables related to the assumed separated representation. The fact that the integrals involved in the formulation can not be written as a sum of products of one-dimensional integrals, this approach reduces the extent of the integration domain.
TL;DR: In this paper, a review of particle filter algorithms for the MEG inverse problem is presented, and the results show encouraging results on both synthetic and experimental MEG data, but the search for a reliable yet general and automatic approach to MEG source modeling is still open.
Abstract: Magnetoencephalography (MEG) is a powerful technique for brain functional studies, which allows investigation of the neural dynamics on a millisecond time-scale. The localization of the neural sources from the measured magnetic fields, based on the solution of an inverse problem, is complicated by several issues. First, the problem is ill-posed: there are infinitely many current distributions explaining a given measurement equally well. Second, the amount of noise on the data is very high, and the main source of noise is the brain itself. Third, the problem is dynamical because the temporal resolution of the data is of the same order of the temporal scale of the neural dynamics. In the last two decades, many different methods have been proposed and applied for solving the MEG inverse problem; however, the search for a reliable yet general and automatic approach to MEG source modeling is still open. Recently we have worked at applying a new class of algorithms, known as particle filters, to the MEG problem. Here we attempt a review of these methods and show encouraging results obtained on both synthetic and experimental MEG data.
TL;DR: In this article, a review of the existing models describing movement of a meteor body in the atmosphere is presented, namely, a theory of a single body and theory of consecutive fragmentation.
Abstract: The paper is devoted to research of movement of meteoric bodies in the terrestrial atmosphere. There is a review of the existing models describing movement, i.e. deceleration, ablation and fragmentation a meteoric body in atmosphere, in the paper beginning; namely a theory of a single body and a theory of consecutive fragmentation. Methods of determination of meteor body parameters by observation data are reviewed. Further the described models and methods have been applied to the analysis of trajectories of several bolides. It is obtained that the model of a single body with the account of ablation describes trajectories of considered bolides with the best accuracy. The trajectory analysis of Benesov’s bolide is carried out, for which there are detailed data of observation. Basic parameters of the bolide are determined, including initial mass. Comparison of the obtained data with results of other authors is made. The second part of the work is devoted to research of interaction of meteoroid fragments in a supersonic flow. We proposed an approximation of numerical data for transverse coefficient by simple analytical function. Further, we obtained analytical solution of a problem on separation of two spherical fragments under the decreasing transverse force without resistance. The new model of layer-by-layer scattering of meteoroid fragments moving as a system of bodies is constructed on the basis of the analytical solutions derived in this work and the numerical data.