Approximate static condensation algorithm for solving multi-material diffusion problems on meshes non-aligned with material interfaces

Kikinzon, Evgeny; Kuznetsov, Yuri A.; Lipnikov, Konstantin; Shashkov, Mikhail

doi:10.1016/j.jcp.2017.06.048

Cited by 15 publications

(10 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Later, Bochev and Hyman [18] extended this work and derived discrete operators such as the discrete wedge product, the discrete codifferential, and the discrete inner products. Robidoux,Hyman,Steinberg and Shashkov,[74,[76][77][78]107,108,111,113,114] used symmetry considerations to construct discretizations on rough grids, within the finite difference/volume setting . In a more recent paper by Robidoux and Steinberg [110] a finite difference discrete vector calculus is presented.…”

Section: Overview Of Mimetic Discretizationsmentioning

confidence: 99%

“…This example is taken from [54,76,78]. The domain is a 2D unit square, Ω = [0, 1] 2 , with 80 shale blocks, Ω s , placed in the domain such that the total area fraction of shale blocks is A shale = 20%, as shown in Figure 10.…”

Section: The Sand-shale Systemmentioning

confidence: 99%

“…The net flux entering the domain (the same as the net flux leaving the domain) is given in Table 1 for varying polynomial degree. A reference value for this solution is given in [54] as 0.5205, and in [78] as 0.519269. In this work the maximum resolution corresponds to 20 × 20 elements and a polynomial degree N = 19, for which the net flux entering the domain is obtained as 0.52010.…”

Section: The Sand-shale Systemmentioning

confidence: 99%

See 2 more Smart Citations

Mimetic Spectral Element Method for Anisotropic Diffusion

Gerritsma

Palha

Jain

et al. 2018

Numerical Methods for PDEs

View full text Add to dashboard Cite

This paper addresses the topological structure of steady, anisotropic, inhomogeneous diffusion problems. Two discrete formulations: a) mixed and b) direct formulations are discussed. Differential operators are represented by sparse incidence matrices, while weighted mass matrices play the role of metric-dependent Hodge matrices. The resulting mixed formulations are point-wise divergence-free if the right hand side function f = 0. The method is inf-sup stable and displays optimal convergence on orthogonal and non-affine grids.

show abstract

Section: Overview Of Mimetic Discretizationsmentioning

confidence: 99%

Section: The Sand-shale Systemmentioning

confidence: 99%

Section: The Sand-shale Systemmentioning

confidence: 99%

See 1 more Smart Citation

Mimetic Spectral Element Method for Anisotropic Diffusion

Gerritsma

Palha

Jain

et al. 2018

Numerical Methods for PDEs

View full text Add to dashboard Cite

show abstract

“…Averaging techniques cannot always faithfully represent the physics of multimaterial cells. Consequently, FLAG employs more sophisticated multimaterial diffusion options, namely the thin mesh [7] and static condensation [8] algorithms. The thin mesh method starts with the volume fractions of each material region, and reconstructs the material interfaces using interface reconstruction methods.…”

Section: Introductionmentioning

confidence: 99%

“…For more details on the algorithm, see Ref. [9]; this method is reported to be second order accurate in Ref. [8].…”

Section: Introductionmentioning

confidence: 99%

Multimaterial Heat Flow Verification

Singleton¹,

Malone²,

L³

2019

Preprint

View full text Add to dashboard Cite

Multimaterial heat diffusion can be a challenging numerical problem when the material boundaries are misaligned with the numerical grid. Even when the boundaries start out aligned, they typically become misaligned through hydrodynamic motion. There are usually a number of methods for handling multimaterial cells in any given hydro code. One of the simplest methods is to replace the multimaterial cell by an average single-material cell whose heat capacity and conductivity are averages over the constituent materials. One can further refine this model by using either the arithmetic or harmonic averages, thereby providing two distinct (albeit naive) multimaterial models for the arithmetic and harmonic averages. More sophisticated models typically involve a surrogate mesh of some kind, as with the thin mesh and static condensation methods. In this paper, we perform rigorous code verification of the multiphysics hydrocode FLAG, including grid resolution studies. We employ a number of newly constructed 2D heat flow solutions that generalize the standard planar sandwich solution, and this paper offers a smorgasbord of exact solutions for heat flow verification. To perform the analyses and to produce the corresponding convergence plots, we employ the code verification tool ExactPack.

show abstract