Voxel‐based fast solution of the Lippmann‐Schwinger equation with smooth material interfaces

Merkert, Dennis; Andrä, Heiko; Kabel, Matthias; Schneider, Matti; Simeon, Bernd

doi:10.1002/pamm.201410277

Cited by 6 publications

(5 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Figure 5 (left) shows the calculated fracture energy G n c relative to the one predicted by Eq. (14). Remember that G c was derived for the rate-independent limit.…”

Section: Effect Of Length Scale Contrast In the Gradient Based Modelmentioning

confidence: 99%

“…planar in 3D) elements cannot be used. Composite pixel based approaches, for example [14], which make use of the interface normal to assign homogenised mechanical properties to grid points near the interfaces, are a step in that direction. But at this point it is not clear how they will perform in situations of damaging interfaces that are inclined or curved.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

FFT-based interface decohesion modelling by a nonlocal interphase

Sharma

Peerlings

Shanthraj

et al. 2018

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

View full text Add to dashboard Cite

In this paper, two nonlocal approaches to incorporate interface damage in fast Fourier transform (FFT) based spectral methods are analysed. In FFT based methods, the discretisation is generally non-conforming to the interfaces and hence interface elements cannot be used. This limitation is remedied using the interfacial band concept, i.e., an interphase region of a finite thickness is used to capture the response of a physical sharp interface. Mesh dependency due to localisation in the softening interphase is avoided by applying established regularisation strategies, integral based nonlocal averaging or gradient based nonlocal damage, which render the interphase nonlocal. Application of these regularisation techniques within the interphase sub-domain in a one dimensional FFT framework is explored. The effectiveness of both approaches in terms of capturing the physical fracture energy, computational aspects and ease of implementation is evaluated. The integral model is found to give more regularised solutions and thus a better approximation of the fracture energy.

show abstract

“…Figure 5 (left) shows the calculated fracture energy G n c relative to the one predicted by Eq. (14). Remember that G c was derived for the rate-independent limit.…”

Section: Effect Of Length Scale Contrast In the Gradient Based Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

FFT-based interface decohesion modelling by a nonlocal interphase

Sharma

Peerlings

Shanthraj

et al. 2018

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

View full text Add to dashboard Cite

show abstract

“…Recently, Willot et al in [35,36] adjusted the integral kernel in the Lippmann-Schwinger equation which led to improved accuracy in approximate solutions, illustrated with a comparison using an analytical solution [37]. A possible improvement can also be achieved by smoothing of material coefficients [38], [11, section 8.3]. Significant attention was granted to improving the linear solvers leading to the accelerated schemes [39,18] or to the Krylov subspace methods, such as conjugate gradients [29,31]; a comparison can be found in [40,41].…”

Section: Fft-based Homogenizationmentioning

confidence: 99%

Improved guaranteed computable bounds on homogenized properties of periodic media by the Fourier-Galerkin method with exact integration

Vondřejc

2016

Int. J. Numer. Meth. Engng

View full text Add to dashboard Cite

Moulinec and Suquet introduced FFT-based homogenization in 1994, and 20 years later, their approach is still effective for evaluating the homogenized properties arising from the periodic cell problem. This paper builds on the author's (2013) variational reformulation approximated by trigonometric polynomials establishing two numerical schemes: Galerkin approximation (Ga) and a version with numerical integration (GaNi). The latter approach, fully equivalent to the original Moulinec-Suquet algorithm, was used to evaluate guaranteed upper-lower bounds on homogenized coefficients incorporating a closed-form doublegrid quadrature. Here, these concepts, based on the primal and dual formulations, are employed for the Ga scheme. For the same computational effort, the Ga outperforms the GaNi with more accurate guaranteed bounds and more predictable numerical behaviors. The quadrature technique leading to block-sparse linear systems is extended here to materials defined via high-resolution images in a way that allows for effective treatment using the FFT. Memory demands are reduced by a reformulation of the double-grid scheme to the original grid scheme using FFT shifts. Minimization of the bounds during iterations of conjugate gradients is effective, particularly when incorporating a solution from a coarser grid. The methodology presented here for the scalar linear elliptic problem could be extended to more complex frameworks.

show abstract

“…This article outgrew from Dennis Merkert's master thesis [29,30], which covered mixing rules for conductivity.…”

Section: Introductionmentioning

confidence: 98%