On the interplay between meshing and discretization in three-dimensional diffusion simulation

Kosik, Robert; Fleischmann, Peter; Haindl, B.; Pietra, Paola; Selberherr, S.

doi:10.1109/43.892848

Cited by 13 publications

(9 citation statements)

References 15 publications

(32 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this case, as mentioned in "Spatial discretizations of the heat Equation" section, the effective-diffusion matrix of the discretization isH =M −1 K. Since each of rowH is proportional to the corresponding row of K [see Eq. (17)]. The following result (see [14]) is valid:…”

Section: On the Issue Of Generating Thermodynamically Compatible Finimentioning

confidence: 92%

“…A natural question to be made is whether the issue of nodal thermodynamic incompatibility of consistent FE Discretizations, discussed here for the 1D and 2D cases, also happens in the 3D case. Recent Finite-Element formulas presented in DMP works [15][16][17][18]) indicate that the problem is also present in 3D FE discretizations.…”

Section: Thermodynamic Incompatibility Of 2d Finite Element Discretizmentioning

confidence: 99%

See 1 more Smart Citation

On the issue that Finite Element discretizations violate, nodally, Clausius’s postulate of the second law of thermodynamics

Limache

Idelsohn

2016

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

View full text Add to dashboard Cite

Discretization processes leading to numerical schemes sometimes produce undesirable effects. One potentially serious problem is that a discretization may produce the loss of validity of some of the physical principles or mathematical properties originally present in the continuous equation. Such loss may lead to uncertain results such as numerical instabilities or unexpected non-physical solutions. As a consequence, the compatibility of a discrete formulation with respect to intrinsic physical principles might be essential for the success of a numerical scheme. This paper addresses such type of issue. Its main objective is to demonstrate that standard Finite Element discretizations of the heat conduction equation violate Clausius's postulate of the second law of thermodynamics, at nodal level. The problem occurs because non-physical, reversed nodal heat-fluxes arise in such discretizations. Conditions for compatibility of discrete nodal heat-fluxes with respect to Clausius's postulate are derived here and named discrete thermodynamic compatibility conditions (DTCC).Simple numerical examples are presented to show the undesirable consequences of such failure. It must be pointed out that such DTCCs have previously appeared in the context of the study of the conditions that make discrete solutions to satisfy the discrete maximum principle (DMP). However, the present article does not put attention on such mathematical principle but on the satisfaction of a fundamental physical one: the second law of thermodynamics. Of course, from the presented point of view, it is clear that the violation of such fundamental law will cause, among different problems, the violation of the DMP.

show abstract

Section: On the Issue Of Generating Thermodynamically Compatible Finimentioning

confidence: 92%

Section: Thermodynamic Incompatibility Of 2d Finite Element Discretizmentioning

confidence: 99%

On the issue that Finite Element discretizations violate, nodally, Clausius’s postulate of the second law of thermodynamics

Limache

Idelsohn

2016

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

View full text Add to dashboard Cite

show abstract

“…Similarly, the authors in [7] discussed the use of such coupling between stabilized finite elements and finite difference time integration on more general problems such as the advection-diffusion-reaction problems. Other related ideas was proposed like, mesh refinement [10], M-matrix theory [11], finite volume method [12], discontinuous Galerkin models [13] and the diffusion-split method [14]. Inspired by all these methods, we believe that the proposed approach works for general meshes, can use any time step or diffusion parameter and, with low computational cost offers a good accuracy order.…”

Section: Introductionmentioning

confidence: 95%

Enriched finite element spaces for transient conduction heat transfer

Hachem¹,

Digonnet²,

Kosseifi³

et al. 2010

Applied Mathematics and Computation

View full text Add to dashboard Cite

“…The Galerkin (standard) version of the finite element method (FEM) applied to diffusion problems, when used with linear tetrahedral (P1) elements, does not in general satisfy the maximum principle [1][2] [3]. Physically, this principle guarantees to obtain the maximum/minimum of the solution only at the initial time or at the boundary, in the latter case a flow from/to the outside must exist [1].…”

Section: Introductionmentioning

confidence: 99%

“…In addition, the geometrical constraints on the mesh prevent from using general Delaunay meshing codes. This leads Kosik et al [3] to directly discourage the use of P1 finite elements, promoting the use of the finite volume method (FVM). However, we can not ignore the wide diffusion of FEM in the existing codes, as well as its versatility compared to FVM.…”

Section: Introductionmentioning

confidence: 99%

A Diffusion-Split Method To Deal With Thermal Shocks Using Standard Linear Tetrahedral Finite Elements

Fachinotti¹

2004

AIP Conference Proceedings

View full text Add to dashboard Cite

The thermal analysis using linear standard tetrahedral finite elements may be affected by spurious local extrema in the regions affected by thermal shocks, in such a severe way to directly discourage the use of these elements. The present work proposes a slight modification to the discrete heat equation in order to obtain a system matrix in Mmatrix form, which assures an oscillation-free solution. The performance of this method is evaluated by means of test case with analytical solution, as well as an industrial application, for which a well-behaved numerical solution is available.

show abstract

On the interplay between meshing and discretization in three-dimensional diffusion simulation

Cited by 13 publications

References 15 publications

On the issue that Finite Element discretizations violate, nodally, Clausius’s postulate of the second law of thermodynamics

On the issue that Finite Element discretizations violate, nodally, Clausius’s postulate of the second law of thermodynamics

Enriched finite element spaces for transient conduction heat transfer

A Diffusion-Split Method To Deal With Thermal Shocks Using Standard Linear Tetrahedral Finite Elements

Contact Info

Product

Resources

About