Discrete Variational Derivative Method

Furihata, Daisuke; Matsuo, Takayasu

doi:10.1201/b10387

Cited by 82 publications

(39 citation statements)

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“…Note that there are the same relations betweenR,Γ ℓ ij , andγ ij . This means that the conformal transformation (7) does not overcome the difficulty of R. Therefore we treat only t to simplify R in this Letter. IfH GR is the function of only t, the density can be expressed as…”

Section: Densitized Adm Formulation Of Only Tmentioning

confidence: 99%

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Constructing of constraint preserving scheme for Einstein equations

Tsuchiya

Yoneda

2017

JSIAM Letters

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We propose a new numerical scheme of evolution for the Einstein equations using the discrete variational derivative method (DVDM). We derive the discrete evolution equation of the constraint using this scheme and show the constraint preserves in the discrete level. In addition, to confirm the numerical stability using this scheme, we perform some numerical simulations by discretized equations with the Crank-Nicolson scheme and with the new scheme, and we find that the new discretized equations have better stability than that of the Crank-Nicolson scheme.

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Section: Densitized Adm Formulation Of Only Tmentioning

confidence: 99%

“…One of the schemes is the discrete variational derivative method (DVDM). DVDM was proposed and has been extended by Furihata, Mori, Matsuo, and Yaguchi [4,5,6,7,8,9], and this method is one of the structure-preserving methods [10]. The method of formulating a discrete system using DVDM is similar to the variational principle.…”

Section: Introductionmentioning

confidence: 99%

Constructing of constraint preserving scheme for Einstein equations

Tsuchiya

Yoneda

2017

JSIAM Letters

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“…Discrete derivatives have been employed in the formulation of conservative implicit methods for classical and solid mechanics. See, among others, [2,9,16,19,22,[41][42][43][44][45][46][47][48]. Although we will not discuss this aspect in the current article, discrete derivatives are also responsible for the preservation of first integrals of the motion, if present in the equations.…”

Section: Discrete Derivativesmentioning

confidence: 99%

“…The formulation of thermodynamically consistent methods follows always the same program: the evolution equations should be discretized using consistent approximations that preserve exactly the key properties behind the relevant conservation laws. In [19], this idea was exploited for general systems of the form (2), revealing that the main tool to attain this goal was to use the discrete derivatives for the energy and entropy densities. In fact, when the thermodynamic variable is chosen to be the entropy density and purely elastic materials, the orthogonality conditions are easily verified for the discrete derivatives, and thermodynamically consistent methods can be constructed in a straightforward manner.…”

Section: Approximating the Discrete Jacobiansmentioning

confidence: 99%

“…The success of structure preserving integrators can be explained by the remarkable properties of the solutions obtained with them when applied to initial (boundary) value problems with special qualitative features [1,2]. By design, certain of these methods have the ability to preserve the symplectic structure of Hamiltonian flows; others, the contractivity in dissipative problems; a third class, the time-reversibility of reversible dynamics; and so on.…”

Section: Introductionmentioning

confidence: 99%

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Energy–entropy–momentum integration schemes for general discrete non‐smooth dissipative problems in thermomechanics

Portillo

Orden

Romero

2017

Numerical Meth Engineering

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We present the theory of novel time-stepping algorithms for general nonlinear, non-smooth, coupled, and thermomechanical problems. The proposed methods are thermodynamically consistent in the sense that their solutions rigorously comply with the two laws of thermodynamics: for isolated systems, they preserve the total energy and the entropy never decreases. Extending previous works on the subject, the newly proposed integrators are applicable to coupled mechanical systems with non-smooth kinetics and can be formulated in arbitrary variables. The ideas are illustrated with a simple non-smooth problem: a rheological model for a thermo-elasto-plastic material with hardening. Numerical simulations verify the qualitative features of the proposed methods and illustrate their excellent numerical stability, which stems precisely from their ability to preserve the structure of the evolution equations they discretize. ENERGY-ENTROPY-MOMENTUM INTEGRATION SCHEMES 777As a result, symplectic, variational, and energy-momentum integrators, among others, have been developed and widely employed in the last decades. As these works show, preserving (part of) the Hamiltonian structure has proven very effective in the numerical solution of stiff models arising in nonlinear mechanics of solids [6,7], flexible and rigid multibody analysis [8,9], contact mechanics [10,11], elastic beams and shells [12][13][14], and so on. For practical applications in solid mechanics, however, all these models fall short because many (if not most) of the interesting problems are not elastic, but rather elastoplastic, or visco-elastic, or exhibit damage, or are coupled with temperature and so on.The class of solids that exhibit some kind of dissipative behavior is much larger than that of elastic solids, and a common mathematical structure for the former is not as apparent as in the Hamiltonian case. This lack of unifying formalism has hindered the formulation of structure preserving methods to nonlinear dissipative phenomena in solids. A few remarkable methods have been proposed in the past for single, specific problems (for example, [15][16][17][18]) but cannot be employed beyond the boundaries of the problem they were designed for.The only class of structure preserving methods for general (smooth) dissipative solids that have been proposed so far is, to the authors' knowledge, the energy-entropy-momentum (EEM) integrators initially proposed in [19] for finite dimensional thermomechanical problems and later extended to the infinite dimensional case [20,21]. In addition to thermoelastic problems, the EEM method has been applied to phase field modeling [22], discrete thermo-visco-plasticity [23], and thermo-visco-elasticity [24][25][26].Energy-entropy-momentum methods could be applied to a broad class of thermomechanical systems after realizing that many of the latter can be formulated as metriplectic models [27]. This mathematical and geometrical formalism generalizes the ideas of Hamiltonian problems and is able to encompass several dissipative phenomen...

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New conservative finite volume element schemes for the modified Korteweg–de Vries equation

Yan

Zhang

2016

Math Methods in App Sciences

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In this paper, three conservative finite volume element schemes are proposed and compared for the modif ied Korteweg–de Vries equation, especially with regard to their accuracy and conservative properties. The schemes are constructed basing on the discrete variational derivative method and the finite volume element method to inherit the properties of the original equation. The theoretical analysis show that three schemes are conservative under suitable boundary conditions as well as unconditionally linear stability. Numerical experiments are given to confirm the theoretical results and the capacity of the proposed methods for capturing the solitary wave phenomena. Copyright © 2016 John Wiley & Sons, Ltd.

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Discrete Variational Derivative Method

Cited by 82 publications

References 0 publications

Constructing of constraint preserving scheme for Einstein equations

Constructing of constraint preserving scheme for Einstein equations

Energy–entropy–momentum integration schemes for general discrete non‐smooth dissipative problems in thermomechanics

New conservative finite volume element schemes for the modified Korteweg–de Vries equation

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