Arbitrary Lagrangian–Eulerian Methods
15 Nov 2004-pp 1-23
TL;DR: In this paper, the authors provide an in-depth survey of arbitrary Lagrangian-Eulerian (ALE) methods, including both conceptual aspects of the mixed kinematical description and numerical implementation details.
Abstract: The aim of the present chapter is to provide an in-depth survey of arbitrary Lagrangian–Eulerian (ALE) methods, including both conceptual aspects of the mixed kinematical description and numerical implementation details. Applications are discussed in fluid dynamics, nonlinear solid mechanics and coupled problems describing fluid–structure interaction. The need for an adequate mesh-update strategy is underlined, and various automatic mesh-displacement prescription algorithms are reviewed. This includes mesh-regularization methods essentially based on geometrical concepts, as well as mesh-adaptation techniques aimed at optimizing the computational mesh according to some error indicator. Emphasis is then placed on particular issues related to the modeling of compressible and incompressible flow and nonlinear solid mechanics problems. This includes the treatment of convective terms in the conservation equations for mass, momentum, and energy, as well as a discussion of stress-update procedures for materials with history-dependent constitutive behavior.
Keywords:
ALE description;
convective transport;
finite elements;
stabilization techniques;
mesh regularization and adaptation;
fluid dynamics;
nonlinear solid mechanics;
stress-update procedures;
fluid–structure interaction
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Book•
24 Feb 2012
TL;DR: This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software.
Abstract: This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.
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TL;DR: This paper develops a geometrically flexible technique for computational fluid-structure interaction (FSI) that directly analyzes a spline-based surface representation of the structure by immersing it into a non-boundary-fitted discretization of the surrounding fluid domain, and introduces the term "immersogeometric analysis" to identify this paradigm.
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TL;DR: In this article, the authors present a new thermomechanical finite element model of ice flow named ISSM (Ice Sheet System Model) that includes higher-order stresses, high spatial resolution capability and data assimilation techniques to better capture ice dynamics and produce realistic simulations of ice sheet flow at the continental scale.
Abstract: Ice flow models used to project the mass balance of ice sheets in Greenland and Antarctica usually rely on the Shallow Ice Approximation (SIA) and the Shallow-Shelf Approximation (SSA), sometimes combined into so-called hybrid models. Such models, while computationally efficient, are based on a simplified set of physical assumptions about the mechanical regime of the ice flow, which does not uniformly apply everywhere on the ice sheet/ice shelf system, especially near grounding lines, where rapid changes are taking place at present. Here, we present a new thermomechanical finite element model of ice flow named ISSM (Ice Sheet System Model) that includes higher-order stresses, high spatial resolution capability and data assimilation techniques to better capture ice dynamics and produce realistic simulations of ice sheet flow at the continental scale. ISSM includes several approximations of the momentum balance equations, ranging from the two-dimensional SSA to the three-dimensional full-Stokes formulation. It also relies on a massively parallelized architecture and state-of-the-art scalable tools. ISSM employs data assimilation techniques, at all levels of approximation of the momentum balance equations, to infer basal drag at the ice-bed interface from satellite radar interferometry-derived observations of ice motion. Following a validation of ISSM with standard benchmarks, we present a demonstration of its capability in the case of the Greenland Ice Sheet. We show ISSM is able to simulate the ice flow of an entire ice sheet realistically at a high spatial resolution, with higher-order physics, thereby providing a pathway for improving projections of ice sheet evolution in a warming climate. Copyright 2012 by the American Geophysical Union.
389 citations
Cites methods from "Arbitrary Lagrangian–Eulerian Metho..."
...We do not use s-coordinates [Pattyn, 2003] but rely on the Arbitrary Lagrangian-Eulerian (ALE) method [Hughes et al., 1981;Donea et al., 2004]....
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TL;DR: In this article, two approaches to model soil-pore fluid coupling in large-deformation analysis using the material point method (MPM) are introduced; one on a model levee failure and the other on a natural cut slope failure (the Selborne experiment conducted by Cooper and co-workers in 1998).
Abstract: Traditional geotechnical analyses for landslides involve failure prediction (i.e. onset of failure) and the design of structures that can safely withstand the applied loads. The analyses provide limited information on the post-failure behaviour. Modern numerical methods are able to simulate large mass movements and there is an opportunity to utilise such methods to evaluate the risks of catastrophic damage if a landslide occurs. In this paper, various large-deformation analysis methods are introduced and their applicability for solving landslide problems is discussed. Since catastrophic landslides often involve seepage forces, consideration of the coupled behaviour of soil and pore fluid is essential. Two approaches to model soil–pore fluid coupling in large-deformation analysis using the material point method (MPM) are introduced. An example simulation is presented for each approach; one on a model levee failure and the other on a natural cut slope failure (the Selborne experiment conducted by Cooper and co-workers in 1998). In the levee failure case, MPM simulation was able to capture a complex failure mechanism including the development of successive shear bands. The simulation was also able to predict excess pore pressure generation during the failure propagation and the subsequent consolidation stage. The simulations demonstrated the importance of the dilation characteristics of soil as well as changes in geometry for the post-failure behaviour. In the Selborne case, MPM was able to simulate the progressive failure of brittle, overconsolidated clay. The evolution of shear stresses along the failure surface was also captured by the MPM. The changes in the pore pressure and the actual shape of the failure surface were simulated by the MPM. The importance of accurately modelling the shear band within the MPM framework is highlighted.
340 citations
TL;DR: It is found that the variational immersed-boundary method for FSI remains robust and effective for heart valve analysis when the background fluid mesh undergoes deformations corresponding to the expansion and contraction of the elastic artery.
Abstract: We propose a framework that combines variational immersed-boundary and arbitrary Lagrangian---Eulerian methods for fluid---structure interaction (FSI) simulation of a bioprosthetic heart valve implanted in an artery that is allowed to deform in the model We find that the variational immersed-boundary method for FSI remains robust and effective for heart valve analysis when the background fluid mesh undergoes deformations corresponding to the expansion and contraction of the elastic artery Furthermore, the computations presented in this work show that the arterial wall deformation contributes significantly to the realism of the simulation results, leading to flow rates and valve motions that more closely resemble those observed in practice
250 citations
References
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Book•
12 Sep 2000
TL;DR: In this paper, the authors present a list of boxes for Lagrangian and Eulerian Finite Elements in One Dimension (LDF) in one dimension, including Beams and Shells.
Abstract: Preface. List of Boxes. Introduction. Lagrangian and Eulerian Finite Elements in One Dimension. Continuum Mechanics. Lagrangian Meshes. Constitutive Models Solution Methods and Stability. Arbitrary Lagrangian Eulerian Formulations. Element Technology. Beams and Shells. Contact--Impact. Appendix 1: Voigt Notation. Appendix 2: Norms. Appendix 3: Element Shape Functions. Glossary. References. Index.
3,928 citations
"Arbitrary Lagrangian–Eulerian Metho..." refers background or methods in this paper
...…description has (momentarily) disappeared from the formulation, so all the concepts, ideas and algorithms of large strain solid mechanics with a Lagrangian description apply, see Bonet and Wood (1997), Belytschko et al. (2000) and Chapter 29 of this Encyclopedia of Computational Mechanics....
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...Constitutive equations for ALE hyperelastoplasticity Hyperelastoplastic models are based on a multiplicative decomposition of the deformation gradient into elastic and plastic parts, F = FeFp, see for instance Belytschko et al. (2000) or Bonet and Wood (1997)....
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...An objective rate of stress is obtained by adding to σ̇ some terms which ensure the objectivity of σ?, see for instance Malvern (1969) or Belytschko et al. (2000)....
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...…time-varying material volume Vt is given by the following well-known expression, often referred to as Reynolds transport theorem (see, for instance, Belytschko et al. (2000) for a detailed proof): d dt ∫ Vt f(x, t) dV = ∫ Vc≡Vt ∂f(x, t) ∂t dV + ∫ Sc≡St f(x, t)v ·n dS (28) which holds for smooth…...
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...Constitutive equations for ALE hypoelastoplasticity Hypoelastoplastic models are based on an additive decomposition of the stretching tensor ∇sv (symmetric part of the velocity gradient) into elastic and plastic parts, see for instance Belytschko et al. (2000) or Bonet and Wood (1997)....
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Book•
01 Jan 1990
TL;DR: In this paper, the authors describe the derivation of conservation laws and apply them to linear systems, including the linear advection equation, the Euler equation, and the Riemann problem.
Abstract: I Mathematical Theory- 1 Introduction- 11 Conservation laws- 12 Applications- 13 Mathematical difficulties- 14 Numerical difficulties- 15 Some references- 2 The Derivation of Conservation Laws- 21 Integral and differential forms- 22 Scalar equations- 23 Diffusion- 3 Scalar Conservation Laws- 31 The linear advection equation- 311 Domain of dependence- 312 Nonsmooth data- 32 Burgers' equation- 33 Shock formation- 34 Weak solutions- 35 The Riemann Problem- 36 Shock speed- 37 Manipulating conservation laws- 38 Entropy conditions- 381 Entropy functions- 4 Some Scalar Examples- 41 Traffic flow- 411 Characteristics and "sound speed"- 42 Two phase flow- 5 Some Nonlinear Systems- 51 The Euler equations- 511 Ideal gas- 512 Entropy- 52 Isentropic flow- 53 Isothermal flow- 54 The shallow water equations- 6 Linear Hyperbolic Systems 58- 61 Characteristic variables- 62 Simple waves- 63 The wave equation- 64 Linearization of nonlinear systems- 641 Sound waves- 65 The Riemann Problem- 651 The phase plane- 7 Shocks and the Hugoniot Locus- 71 The Hugoniot locus- 72 Solution of the Riemann problem- 721 Riemann problems with no solution- 73 Genuine nonlinearity- 74 The Lax entropy condition- 75 Linear degeneracy- 76 The Riemann problem- 8 Rarefaction Waves and Integral Curves- 81 Integral curves- 82 Rarefaction waves- 83 General solution of the Riemann problem- 84 Shock collisions- 9 The Riemann problem for the Euler equations- 91 Contact discontinuities- 92 Solution to the Riemann problem- II Numerical Methods- 10 Numerical Methods for Linear Equations- 101 The global error and convergence- 102 Norms- 103 Local truncation error- 104 Stability- 105 The Lax Equivalence Theorem- 106 The CFL condition- 107 Upwind methods- 11 Computing Discontinuous Solutions- 111 Modified equations- 1111 First order methods and diffusion- 1112 Second order methods and dispersion- 112 Accuracy- 12 Conservative Methods for Nonlinear Problems- 121 Conservative methods- 122 Consistency- 123 Discrete conservation- 124 The Lax-Wendroff Theorem- 125 The entropy condition- 13 Godunov's Method- 131 The Courant-Isaacson-Rees method- 132 Godunov's method- 133 Linear systems- 134 The entropy condition- 135 Scalar conservation laws- 14 Approximate Riemann Solvers- 141 General theory- 1411 The entropy condition- 1412 Modified conservation laws- 142 Roe's approximate Riemann solver- 1421 The numerical flux function for Roe's solver- 1422 A sonic entropy fix- 1423 The scalar case- 1424 A Roe matrix for isothermal flow- 15 Nonlinear Stability- 151 Convergence notions- 152 Compactness- 153 Total variation stability- 154 Total variation diminishing methods- 155 Monotonicity preserving methods- 156 l1-contracting numerical methods- 157 Monotone methods- 16 High Resolution Methods- 161 Artificial Viscosity- 162 Flux-limiter methods- 1621 Linear systems- 163 Slope-limiter methods- 1631 Linear Systems- 1632 Nonlinear scalar equations- 1633 Nonlinear Systems- 17 Semi-discrete Methods- 171 Evolution equations for the cell averages- 172 Spatial accuracy- 173 Reconstruction by primitive functions- 174 ENO schemes- 18 Multidimensional Problems- 181 Semi-discrete methods- 182 Splitting methods- 183 TVD Methods- 184 Multidimensional approaches
3,827 citations
Book•
01 Jan 1969
TL;DR: In this article, the authors propose a linearized theory of elasticity for tensors, which they call Linearized Theory of Elasticity (LTHE), which is based on tensors and elasticity.
Abstract: 1. Vectors and Tensors. 2. Strain and Deformation. 3. General Principles. 4. Constitutive Equations. 5. Fluid Mechanics. 6. Linearized Theory of Elasticity. Appendix I: Tensors. Appendix II: Orthogonal Curvilinear.
3,658 citations
"Arbitrary Lagrangian–Eulerian Metho..." refers background or methods in this paper
...The algorithms of continuum mechanics usually make use of two classical descriptions of motion: the Lagrangian description and the Eulerian description, see, for instance, Malvern (1969)....
[...]
...An objective rate of stress is obtained by adding to σ̇ some terms which ensure the objectivity of σ?, see for instance Malvern (1969) or Belytschko et al. (2000)....
[...]
TL;DR: The term immersed boundary (IB) method is used to encompass all such methods that simulate viscous flows with immersed (or embedded) boundaries on grids that do not conform to the shape of these boundaries.
Abstract: The term “immersed boundary method” was first used in reference to a method developed by Peskin (1972) to simulate cardiac mechanics and associated blood flow. The distinguishing feature of this method was that the entire simulation was carried out on a Cartesian grid, which did not conform to the geometry of the heart, and a novel procedure was formulated for imposing the effect of the immersed boundary (IB) on the flow. Since Peskin introduced this method, numerous modifications and refinements have been proposed and a number of variants of this approach now exist. In addition, there is another class of methods, usually referred to as “Cartesian grid methods,” which were originally developed for simulating inviscid flows with complex embedded solid boundaries on Cartesian grids (Berger & Aftosmis 1998, Clarke et al. 1986, Zeeuw & Powell 1991). These methods have been extended to simulate unsteady viscous flows (Udaykumar et al. 1996, Ye et al. 1999) and thus have capabilities similar to those of IB methods. In this review, we use the term immersed boundary (IB) method to encompass all such methods that simulate viscous flows with immersed (or embedded) boundaries on grids that do not conform to the shape of these boundaries. Furthermore, this review focuses mainly on IB methods for flows with immersed solid boundaries. Application of these and related methods to problems with liquid-liquid and liquid-gas boundaries was covered in previous reviews by Anderson et al. (1998) and Scardovelli & Zaleski (1999). Consider the simulation of flow past a solid body shown in Figure 1a. The conventional approach to this would employ structured or unstructured grids that conform to the body. Generating these grids proceeds in two sequential steps. First, a surface grid covering the boundaries b is generated. This is then used as a boundary condition to generate a grid in the volume f occupied by the fluid. If a finite-difference method is employed on a structured grid, then the differential form of the governing equations is transformed to a curvilinear coordinate system aligned with the grid lines (Ferziger & Peric 1996). Because the grid conforms to the surface of the body, the transformed equations can then be discretized in the
3,184 citations