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Lin Lin - One of the best experts on this subject based on the ideXlab platform.
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Variational Structure of luttinger ward formalism and bold diagrammatic expansion for euclidean lattice field theory
Proceedings of the National Academy of Sciences of the United States of America, 2018Co-Authors: Lin Lin, Michael A LindseyAbstract:The Luttinger-Ward functional was proposed more than five decades ago and has been used to formally justify most practically used Green's function methods for quantum many-body systems. Nonetheless, the very existence of the Luttinger-Ward functional has been challenged by recent theoretical and numerical evidence. We provide a rigorously justified Luttinger-Ward formalism, in the context of Euclidean lattice field theory. Using the Luttinger-Ward functional, the free energy can be Variationally minimized with respect to Green's functions in its domain. We then derive the widely used bold diagrammatic expansion rigorously, without relying on formal arguments such as partial resummation of bare diagrams to infinite order.
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Variational Structure of luttinger ward formalism and bold diagrammatic expansion for euclidean lattice field theory
arXiv: Mathematical Physics, 2017Co-Authors: Lin Lin, Michael A LindseyAbstract:The Luttinger-Ward functional was proposed more than five decades ago to provide a link between static and dynamic quantities in a quantum many-body system. Despite its widespread usage, the derivation of the Luttinger-Ward functional remains valid only in the formal sense, and even the very existence of this functional has been challenged by recent numerical evidence. In a simpler and yet highly relevant regime, namely the Euclidean lattice field theory, we rigorously prove that the Luttinger-Ward functional is a well-defined universal functional over all physical Green's functions. Using the Luttinger-Ward functional, the free energy can be Variationally minimized with respect to Green's functions in its domain. We then derive the widely used bold diagrammatic expansion rigorously, without relying on formal arguments such as partial resummation of bare diagrams to infinite order.
Christian Miehe - One of the best experts on this subject based on the ideXlab platform.
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aspects of finite element formulations for the coupled problem of poroelasticity based on a canonical minimization principle
Computational Mechanics, 2019Co-Authors: Stephan Teichtmeister, Steffen Mauthe, Christian MieheAbstract:This work presents a new finite element treatment of the coupled problem of Darcy–Biot-type fluid transport in porous media undergoing large deformations, that is free from any stabilization techniques. The formulation bases on an incremental two-field minimization principle that is constrained by the equation of continuity for the fluid mass content and determines at a given state the deformation and the fluid mass flux vector. The big advantage of the minimization formulation over classical saddle point principles of poroelasticity is the omission of the inf-sup condition—a condition that makes the construction of stable and computationally efficient finite element formulations difficult. Due to the $$H(\hbox {Div}, {\mathcal B}_0)$$ Variational Structure of the minimization principle on the fluid side, lowest order Raviart–Thomas elements are used for the conforming approximation of the fluid mass flux. Furthermore, a standard nodal-based element using bilinear interpolation for both fields combined with a reduced numerical integration of the (volumetric) coupling term is analyzed and used for the solution of the minimization principle. Representative numerical examples demonstrate the performance of the proposed finite element designs of the minimization principle and clearly underline advantages over finite element formulations of the classical two-field saddle point principle formulated in deformation and fluid potential.
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mixed Variational principles and robust finite element implementations of gradient plasticity at small strains
International Journal for Numerical Methods in Engineering, 2013Co-Authors: Christian Miehe, Fadi Aldakheel, Steffen MautheAbstract:SUMMARY This work outlines a theoretical and computational framework of gradient plasticity based on a rigorous exploitation of mixed Variational principles. In contrast to classical local approaches to plasticity based on locally evolving internal variables, order parameter fields are taken into account governed by additional balance-type PDEs including micro-structural boundary conditions. This incorporates non-local plastic effects based on length scales, which reflect properties of the material micro-Structure. We develop a unified Variational framework based on mixed saddle point principles for the evolution problem of gradient plasticity, which is outlined for the simple model problem of von Mises plasticity with gradient-extended hardening/softening response. The mixed Variational Structure includes the hardening/softening variable itself as well as its dual driving force. The numerical implementation exploits the underlying Variational Structure, yielding a canonical symmetric Structure of the monolithic problem. It results in a novel finite element (FE) design of the coupled problem incorporating a long-range hardening/softening parameter and its dual driving force. This allows a straightforward local definition of plastic loading-unloading driven by the long-range fields, providing very robust FE implementations of gradient plasticity. This includes a rational method for the definition of elastic-plastic-boundaries in gradient plasticity along with a post-processor that defines the plastic variables in the elastic range. We discuss alternative mixed FE designs of the coupled problem, including a local-global solution strategy of short-range and long-range fields. This includes several new aspects, such as extended Q1P0-type and Mini-type finite elements for gradient plasticity. All methods are derived in a rigorous format from Variational principles. Numerical benchmarks address advantages and disadvantages of alternative FE designs, and provide a guide for the evaluation of simple and robust schemes for Variational gradient plasticity. Copyright © 2013 John Wiley & Sons, Ltd.
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a robust algorithm for configurational force driven brittle crack propagation with r adaptive mesh alignment
International Journal for Numerical Methods in Engineering, 2007Co-Authors: Christian Miehe, Ercan GursesAbstract:The paper considers a Variational formulation of brittle fracture in elastic solids and proposes a numerical implementation by a finite element method. On the theoretical side, we outline a consistent thermodynamic framework for crack propagation in an elastic solid. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius–Planck inequality in the sense of Coleman's method. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip and minimizes the incremental energy release. On the numerical side, we exploit this Variational Structure in terms of crack-driving configurational forces. First, a standard finite element discretization in space yields a discrete formulation of the global dissipation in terms configurational nodal forces. As a consequence, the constitutive setting of crack propagation in the space-discretized finite element context is naturally related to discrete nodes of a typical finite element mesh. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity is performed by the doubling of critical nodes and interface segments of the mesh. Critical for the success of this procedure is its embedding into an r-adaptive crack-segment reorientation procedure with configurational-force-based directional indicator. Here, successive crack releases appear in discrete steps associated with the given space discretization. These are performed by a staggered loading–release algorithm of energy minimization at frozen crack state followed by the successive crack releases at frozen deformation. This constitutes a sequence of positive-definite discrete subproblems with successively decreasing overall stiffness, providing an extremely robust algorithmic setting in the postcritical range. We demonstrate the performance of the formulation by means of representative numerical simulations. Copyright © 2007 John Wiley & Sons, Ltd.
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on multiscale fe analyses of heterogeneous Structures from homogenization to multigrid solvers
International Journal for Numerical Methods in Engineering, 2007Co-Authors: Christian Miehe, C G BayreutherAbstract:Heterogeneous Structures like composites often need a fine-scale resolution of micro-effects which influence the macroscopic overall response. This is of particular relevance in the fully non-linear range of large strains and inelastic material response of the constituents. Suitable solution methods introduce a multifield scenario of hierarchically superimposed states on different length scales. For big differences of micro- and macro-scales, the argument of scale separation induces the application of homogenization methods. Such types of physical multiscale approaches can be treated by nested multilevel finite element analyses that discretize both the fine-scale micro-Structure as well as the macroscopic boundary-value problem. In contrast, small-scale differences require full resolution of the heterogeneous Structure. Effective solution methods for the resulting large-scale problems with strongly oscillating properties are suitably designed geometric multigrid techniques, which may be considered as numerical multiscale approaches. In both scenarios, a key ingredient is the suitable formulation of scale bridging algorithms that govern the transfer between different scales. The paper outlines new mesh-bridging techniques in a deformation-driven context for fully non-linear response, which exploit in a non-trivial manner weak constraints on the average deformation in typical finite element patches. The framework is based on an incremental Variational Structure of finite inelasticity. The proposed new formulations provide Variational-based homogenization algorithms for physical multiscale scenarios and problem-dependent optimal finite element grid transfers for numerical multiscale scenarios of heterogeneous materials. Copyright © 2007 John Wiley & Sons, Ltd.
Michael A Lindsey - One of the best experts on this subject based on the ideXlab platform.
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Variational Structure of luttinger ward formalism and bold diagrammatic expansion for euclidean lattice field theory
Proceedings of the National Academy of Sciences of the United States of America, 2018Co-Authors: Lin Lin, Michael A LindseyAbstract:The Luttinger-Ward functional was proposed more than five decades ago and has been used to formally justify most practically used Green's function methods for quantum many-body systems. Nonetheless, the very existence of the Luttinger-Ward functional has been challenged by recent theoretical and numerical evidence. We provide a rigorously justified Luttinger-Ward formalism, in the context of Euclidean lattice field theory. Using the Luttinger-Ward functional, the free energy can be Variationally minimized with respect to Green's functions in its domain. We then derive the widely used bold diagrammatic expansion rigorously, without relying on formal arguments such as partial resummation of bare diagrams to infinite order.
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Variational Structure of luttinger ward formalism and bold diagrammatic expansion for euclidean lattice field theory
arXiv: Mathematical Physics, 2017Co-Authors: Lin Lin, Michael A LindseyAbstract:The Luttinger-Ward functional was proposed more than five decades ago to provide a link between static and dynamic quantities in a quantum many-body system. Despite its widespread usage, the derivation of the Luttinger-Ward functional remains valid only in the formal sense, and even the very existence of this functional has been challenged by recent numerical evidence. In a simpler and yet highly relevant regime, namely the Euclidean lattice field theory, we rigorously prove that the Luttinger-Ward functional is a well-defined universal functional over all physical Green's functions. Using the Luttinger-Ward functional, the free energy can be Variationally minimized with respect to Green's functions in its domain. We then derive the widely used bold diagrammatic expansion rigorously, without relying on formal arguments such as partial resummation of bare diagrams to infinite order.
Qi Wang - One of the best experts on this subject based on the ideXlab platform.
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a second order fully discrete linear energy stable scheme for a binary compressible viscous fluid model
Journal of Computational Physics, 2019Co-Authors: Xueping Zhao, Qi WangAbstract:Abstract We present a linear, second order fully discrete numerical scheme on a staggered grid for a thermodynamically consistent hydrodynamic phase field model of binary compressible fluid flows, derived from the generalized Onsager Principle. The hydrodynamic model possesses not only the Variational Structure in its constitutive equation, but also warrants the mass, linear momentum conservation as well as energy dissipation. We first reformulate the model using the energy quadratization method into an equivalent form and then discretize the reformulated model to obtain a semidiscrete partial differential equation system using the Crank-Nicolson method in time. The semi-discrete numerical scheme preserves the mass conservation and energy dissipation law in time. Then, we discretize the semi-discrete PDE system on a staggered grid in space to arrive at a fully discrete scheme using 2nd order finite difference methods, which respects a discrete energy dissipation law. We prove the unique solvability of the linear system resulting from the fully discrete scheme. Mesh refinements and numerical examples on phase separation due to spinodal decomposition in binary polymeric fluids and interface evolution in the gas-liquid mixture are presented to show the convergence property and the usefulness of the new scheme in applications, respectively.
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an energy stable algorithm for a quasi incompressible hydrodynamic phase field model of viscous fluid mixtures with variable densities and viscosities
Computer Physics Communications, 2017Co-Authors: Yuezheng Gong, Qi Wang, Jia ZhaoAbstract:Abstract A quasi-incompressible hydrodynamic phase field model for flows of fluid mixtures of two incompressible viscous fluids of distinct densities and viscosities is derived by using the generalized Onsager principle, which warrants the Variational Structure, the mass conservation and energy dissipation law. We recast the model in an equivalent form and discretize the equivalent system in space firstly to arrive at a time-dependent ordinary differential and algebraic equation (DAE) system, which preserves the mass conservation and energy dissipation law at the semi-discrete level. Then, we develop a temporal discretization scheme for the DAE system, where the mass conservation and the energy dissipation law are once again preserved at the fully discretized level. We prove that the fully discretized algorithm is unconditionally energy stable. Several numerical examples, including drop dynamics of viscous fluid drops immersed in another viscous fluid matrix and mixing dynamics of binary polymeric solutions, are presented to show the convergence property as well as the accuracy and efficiency of the new scheme.
Giorgio Fusco - One of the best experts on this subject based on the ideXlab platform.
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a maximum principle for systems with Variational Structure and an application to standing waves
Journal of the European Mathematical Society, 2015Co-Authors: Nicholas D. Alikakos, Giorgio FuscoAbstract:We establish via Variational methods the existence of a standing wave together with an estimate on the convergence to its asymptotic states for a bistable system of partial differential equations on a periodic domain. The main tool is a replacement lemma which has as a corollary a maximum principle for local minimizers.
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Entire Solutions to Equivariant Elliptic Systems with Variational Structure
Archive for Rational Mechanics and Analysis, 2011Co-Authors: Nicholas D. Alikakos, Giorgio FuscoAbstract:We consider the system Δ u − W _ u ( u ) = 0, where $${u : \mathbb{R}^n \to \mathbb{R}^n}$$ , for a class of potentials $${W : \mathbb{R}^n \to \mathbb{R}}$$ that possess several global minima and are invariant under a general finite reflection group G . We establish existence of nontrivial G -equivariant entire solutions connecting the global minima of W along certain directions at infinity.
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entire solutions to equivariant elliptic systems with Variational Structure
arXiv: Analysis of PDEs, 2008Co-Authors: Nicholas D. Alikakos, Giorgio FuscoAbstract:In the present paper we consider the system {\Delta}u - W_u (u) = 0, where u: R^n to R^n, for a class of potentials W: R^n to R that possess several global minima and are invariant under a general finite reflection group G. We establish existence of nontrivial entire solutions connecting the global minima of W along certain directions at infinity.