Computational first-order homogenization in chemo-mechanics

Kaessmair, Stefan; Steinmann, Paul

doi:10.1007/s00419-017-1287-0

Cited by 18 publications

(24 citation statements)

References 30 publications

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“…First, the formulation considering the concentration and the displacement (strain) as the primary field variables is presented, which requires C 1continuity and is therefore cumbersome to implement numerically. Next, using a Legendre transform, the primal field variables are transformed to the chemical potential and strain [24]. This formulation requires only C 0 -continuity and standard finite elements can be used for the implementation.…”

Section: Coupled Diffusion-mechanics Formulationmentioning

confidence: 99%

“…Equation (16), however, involves the third-order derivative of u and its numerical solution therefore requires a C 1 -continuous finite element formulation. Various other solution techniques have also been proposed in the literature for this type of problems, see for example [24,34]. In the current work, following [24], a Legendre transform is performed on the Helmholtz's free energy density function ψ(c, ε) to obtain a dual energy density function ω, for which the primary field variables are the chemical potential μ and the strain ε.…”

Section: (C ε) Formulationmentioning

confidence: 99%

“…Analytical methods for the solution of coupled diffusion-mechanics problems are limited to simple geometrical shapes [22], therefore approximate solutions using numerical techniques such as finite elements are often required [17]. However, with a complex microstructures [23] and transient phenomena [24], the direct numerical simulations (DNS) become prohibitively expensive.…”

Section: Introductionmentioning

confidence: 99%

“…The effective behavior is computed from a representative microscopic element (RVE) [27] and transferred to the macroscale. The computational homogenization of transient phenomena, as associated with lithium diffusion in batteries, has been the focus of research recently [24,28]. Effective responses have to be computed at each macroscopic material point at each time step, making homogenization of transient phenomena computationally demanding.…”

Section: Introductionmentioning

confidence: 99%

“…In this work, we propose a computationally efficient method for the homogenization of coupled diffusion-mechanics for the cathode material of a lithium ion battery. The homogenization of the underlying diffusion and mechanical problems is performed separately by using the method proposed in [24]. For the diffusion problem, equivalence of virtual power (extended Hill-Mandel condition) is considered, while for the mechanical problem equivalence of virtual work is used (standard Hill-Mandel).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Enriched continuum for multi-scale transient diffusion coupled to mechanics

Waseem

Heuzé

Stainier

et al. 2020

Adv. Model. and Simul. in Eng. Sci.

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In this article, we present a computationally efficient homogenization technique for linear coupled diffusion-mechanics problems. It considers a linear chemo-mechanical material model at the fine scale, and relies on a full separation of scales between the time scales governing diffusion and mechanical phenomena, and a relaxed separation of scales for diffusion between the matrix and the inclusion. When the characteristic time scales associated with mass diffusion are large compared to those linked to the deformation, the mechanical problem can be considered to be quasi-static, and a full separation of scales can be assumed, whereas the diffusion problem remains transient. Using equivalence of the sum of virtual powers of internal and transient forces between the microscale and the macroscale, a homogenization framework is derived for the mass diffusion, while for the mechanical case, considering its quasi-static nature, the classical equivalence of the virtual work of internal forces is used instead. Model reduction is then applied at the microscale. Assuming a relaxed separation of scales for diffusion phenomena, the microscopic fields are split into steady-state and transient parts, for which distinct reduced bases are extracted, using static condensation for the steady-state part and the solution of an eigenvalue problem for the transient part. The model reduction at the microscale results in emergent macroscopic enriched field variables, evolution of which is described with a set of ordinary differential equations which are inexpensive to solve. The net result is a coupled diffusion-mechanics enriched continuum at the macroscale. Numerical examples are conducted for the cathode-electrolyte system characteristic of a lithium ion battery. The proposed reduced order homogenization method is shown to be able to capture the coupled behavior of this system, whereby high computational gains are obtained relative to a full computational homogenization method.

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Section: Coupled Diffusion-mechanics Formulationmentioning

confidence: 99%

Section: (C ε) Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Enriched continuum for multi-scale transient diffusion coupled to mechanics

Waseem

Heuzé

Stainier

et al. 2020

Adv. Model. and Simul. in Eng. Sci.

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Variationally consistent computational homogenization of chemomechanical problems with stabilized weakly periodic boundary conditions

Kaessmair

Runesson

Steinmann

et al. 2021

Numerical Meth Engineering

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A variationally consistent model-based computational homogenization approach for transient chemomechanically coupled problems is developed based on the classical assumption of first-order prolongation of the displacement, chemical potential, and (ion) concentration fields within a representative volume element (RVE). The presence of the chemical potential and the concentration as primary global fields represents a mixed formulation, which has definite advantages. Nonstandard diffusion, governed by a Cahn-Hilliard type of gradient model, is considered under the restriction of miscibility. Weakly periodic boundary conditions on the pertinent fields provide the general variational setting for the uniquely solvable RVE-problem(s). These boundary conditions are introduced with a novel approach in order to control the stability of the boundary discretization, thereby circumventing the need to satisfy the LBB-condition: the penalty stabilized Lagrange multiplier formulation, which enforces stability at the cost of an additional Lagrange multiplier for each weakly periodic field (three fields for the current problem). In particular, a neat result is that the classical Neumann boundary condition is obtained when the penalty becomes very large. In the numerical examples, we investigate the following characteristics: the mesh convergence for different boundary approximations, the sensitivity for the choice of penalty parameter, and the influence of RVE-size on the macroscopic response.

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On the Computational Homogenization of Deformation–Diffusion Processes

Polukhov

Keip

2021

Proc Appl Math and Mech

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In various industrial applications, materials are usually considered in the form of composites in order to take advantage of further enhanced physical properties, particularly by designing complex microstructures. Therefore, it is of high interest to computationally model as well as predict the response of not only elastic materials but also materials showing characteristic coupling phenomena. In the present contribution, we are considering the computational homogenization of deformationdiffusion processes (see also [1, 4]) in a minimization-based formulation (see [2, 3, 6]). In this approach, the primary fields are the rate of the deformation map and fluid volume flux which is incorporated in a rate-type variational principle. The timediscrete version of the problem is implemented into a conforming Raviart-Thomas-type finite element formulation. Finally, we present numerical examples to show further aspects of the formulation.

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Computational first-order homogenization in chemo-mechanics

Cited by 18 publications

References 30 publications

Enriched continuum for multi-scale transient diffusion coupled to mechanics

Enriched continuum for multi-scale transient diffusion coupled to mechanics

Variationally consistent computational homogenization of chemomechanical problems with stabilized weakly periodic boundary conditions

On the Computational Homogenization of Deformation–Diffusion Processes

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