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Comparison of classical and higher order continuum models for shear failure of concrete
Published in Günther Meschke, Bernhard Pichler, Jan G. Rots, Computational Modelling of Concrete and Concrete Structures, 2022
P. Hofer, M. Neuner, G. Hofstetter
The three investigated models were implemented in Abaqus/Standard (Abaqus 2015) using the respective UMAT (CDP model) and UEL (GCDP and GMCDP models) interfaces. For the GCDP and GMCDP models, user defined finite elements are employed, as fully coupled problems are being solved. All models were implemented in the Marmot material modeling toolbox library (Dummer, Mader, Neuner, & Schreter 2021) using C++ programming language. For achieving a quadratic rate of convergence in nonlinear simulations, the consistent tangent operators are computed for all models. To this end, the Eigen template library (Guennebaud et al. 2010) is used for linear algebra computations for the CDP and the GCDP models, while for the GMCDP model tensor contraction operations are performed using the Fastor library (Poya, Gil, & Ortigosa 2017) at material and finite element level.
Continuum and Atomic-Scale Finite Element Modeling of Multilayer Self-Positioning Nanostructures
Published in Sarhan M. Musa, Computational Finite Element Methods in Nanotechnology, 2013
The expansion operation should be accompanied by filling symmetrical terms of the stress tensor σij = σji. For the elasticity tensor, contraction and expansion operations are carried out in the following ways: () Cpq=Cmpnpmqnq,p=1…6,q=1…6,Cmpnpmqnq,=Cpq,p=1…6,q=1…6.
On extending and optimising the direct product decomposition
Published in Molecular Physics, 2019
The original formulation and implementation of the DPD only considered 4-dimensional (and by use of a dummy index, 3-dimensional) tensors. The concept may be readily extended to higher (and lower) dimensionality however. For a generic tensor contraction we may split the operation into g sub-contractions, one for each combination of (given one irrep the others are fixed by any two of the conditions). Within each group, the indices and their irrep combinations such as may be linearised in any fashion. However, we may in particular consider a recursive application of the DPD. For example, consider a tensor decomposed as When linearising the abcd index group, we may decompose it into g sub-groups based on the combinations of as for a 4-dimensional tensor. These sub-groups are then trivially linearised by decomposition into etc.
A Port-Hamiltonian formulation of linear thermoelasticity and its mixed finite element discretization
Published in Journal of Thermal Stresses, 2021
A. Brugnoli, D. Alazard, V. Pommier-Budinger, D. Matignon
Operator grad is the gradient of a scalar field, while div is the divergence of a vector field. The notation denotes the tensor contraction. The reader may consult [19, Chapter 1] or [20, Chapter 8] for a detailed derivation on these equations.