A conservative penalisation strategy for the semi-implicit time discretisation of the incompressible elastodynamics equation

We propose a generalisation of the Mullins–Sekerka problem to model phase separation in multi-component systems. The model includes equilibrium equations in bulk, the Gibbs–Thomson relation on the interfaces, Young's law at triple junctions, together with a dynamic law of Stefan type. Using formal asymptotic expansions, we establish the relationship to a transition layer model known as the Cahn-Hilliard system. We introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds, together with measure theoretical tools. Through an implicit time discretisation, we construct approximate solutions by stepwise minimisation. Under the assumption that there is no loss of area as the time step tends to zero, we show the existence of a weak solution.

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Staggered explicit-implicit time-discretisation for elastodynamics with dissipative internal variables

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2020040 ◽

2020 ◽

Author(s):

Tomas Roubicek ◽

Chrysoula Tsogka

Keyword(s):

A Priori Estimates ◽

A Priori ◽

Time Discretization ◽

Small Strain ◽

Implicit Time ◽

Internal Variables ◽

Spatial Gradients ◽

Poroelastic Media ◽

Time Discretisation ◽

Space Formulation

An extension of the two-step staggered time discretization of linear elastodynamics in stress-velocity form to systems involving internal variables subjected to a possibly non-linear dissipative evolution is proposed. The original scheme is thus enhanced by another step for the internal variables which, in general, is implicit, although even this step might be explicit if no spatial gradients of the internal variables are involved. Using an abstract Banach-space formulation, a-priori estimates and convergence are proved under a CFL condition. The developed three-step scheme finds applications in various problems of continuum mechanics at small strain. Here, we consider in particular plasticity, viscoelasticity (creep), diffusion in poroelastic media, and damage.

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Comparing various influences on adhesive contact with friction

Selected Scientific Papers - Journal of Civil Engineering ◽

10.1515/sspjce-2019-0013 ◽

2019 ◽

Vol 14 (2) ◽

pp. 7-18

Author(s):

Roman Vodička

Keyword(s):

Boundary Element Method ◽

Coulomb Friction ◽

Frictional Contact ◽

Adhesive Contact ◽

Implicit Time ◽

Engineering Structures ◽

Time Discretisation ◽

Elastic Bodies ◽

Internal Parameters ◽

Programming Algorithms

AbstractA general computational model covering many types of frictional contact interfaces between visco-elastic bodies is considered for some cases physically relevant in numerical analysis of contact in civil engineering structures. The relations between mechanical quantities and internal parameters of the model are illustrated in a couple of simplified examples including cohesive contact combined with Coulomb friction and/or interface plasticity. The computations are implemented a semi-implicit time discretisation, quadratic programming algorithms, and the boundary-element method.

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Solution to the Compressible Navier-Stokes Equations of Motion by Chebyshev Polynomials with Implicit Time Stepping

10.21236/ada194119 ◽

1988 ◽

Author(s):

Leonidas Sakell

Keyword(s):

Chebyshev Polynomials ◽

Stokes Equations ◽

Equations Of Motion ◽

Navier Stokes ◽

Implicit Time ◽

Navier Stokes Equations ◽

Time Stepping

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Semi-Implicit Time Integration in Limited Area Model

10.21236/ada226818 ◽

1989 ◽

Author(s):

Isidore Halberstam

Keyword(s):

Time Integration ◽

Implicit Time ◽

Limited Area ◽

Limited Area Model ◽

Area Model ◽

Implicit Time Integration

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Modeling Structural Dynamics Using FE-Meshfree QUAD4 Element with Radial-Polynomial Basis Functions

Materials ◽

10.3390/ma14092288 ◽

2021 ◽

Vol 14 (9) ◽

pp. 2288

Author(s):

Hongming Luo ◽

Guanhua Sun

Keyword(s):

Structural Dynamics ◽

Interpolation Method ◽

Mass Matrix ◽

Time Integration ◽

Partition Of Unity ◽

Implicit Time ◽

Lumped Mass ◽

Point Interpolation Method ◽

Point Interpolation ◽

Consistent Mass Matrix

The PU (partition-of-unity) based FE-RPIM QUAD4 (4-node quadrilateral) element was proposed for statics problems. In this element, hybrid shape functions are constructed through multiplying QUAD4 shape function with radial point interpolation method (RPIM). In the present work, the FE-RPIM QUAD4 element is further applied for structural dynamics. Numerical examples regarding to free and forced vibration analyses are presented. The numerical results show that: (1) If CMM (consistent mass matrix) is employed, the FE-RPIM QUAD4 element has better performance than QUAD4 element under both regular and distorted meshes; (2) The DLMM (diagonally lumped mass matrix) can supersede the CMM in the context of the FE-RPIM QUAD4 element even for the scheme of implicit time integration.

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On Hydromechanical Interaction during Propagation of Localized Damage in Rocks

Minerals ◽

10.3390/min11020162 ◽

2021 ◽

Vol 11 (2) ◽

pp. 162

Author(s):

A.A. Jameei ◽

S. Pietruszczak

Keyword(s):

Numerical Study ◽

Fluid Pressure ◽

Implicit Time ◽

Integration Scheme ◽

Internal Length ◽

Element Analysis ◽

Equivalent Permeability ◽

Mechanical Coupling ◽

Splitting Test ◽

Localized Damage

This paper provides a mathematical description of hydromechanical coupling associated with propagation of localized damage. The framework incorporates an embedded discontinuity approach and addresses the assessment of both hydraulic and mechanical properties in the region intercepted by a fracture. Within this approach, an internal length scale parameter is explicitly employed in the definition of equivalent permeability as well as the tangential stiffness operators. The effect of the progressive evolution of damage on the hydro-mechanical coupling is examined and an evolution law is derived governing the variation of equivalent permeability with the continuing deformation. The framework is verified by a numerical study involving 3D simulation of an axial splitting test carried out on a saturated sample under displacement and fluid pressure-controlled conditions. The finite element analysis incorporates the Polynomial-Pressure-Projection (PPP) stabilization technique and a fully implicit time integration scheme.

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