Flux Correction Tools for Finite Elements

Kuzmin, Dmitri; Turek, Stefan

doi:10.1006/jcph.2001.6955

Cited by 171 publications

(242 citation statements)

References 27 publications

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“…It turns out that our scheme provides a much sharper resolution in comparison with the second-order high-resolution scheme of Osher and Sethian [23]. Let us note that in contrast to the classical works of Boris, Book and their collaborators, we derive the essential information for our algorithms on the discrete basis, while compared to the approach of Kuzmin and Turek [18] our proceeding is technically relatively simple. Furthermore, both mentioned FCT approaches rely on an underlying additive splitting of the backward diffusion into fluxes between computational nodes: especially in the multidimensional case, the mentioned works proceed along the considerations of Zalesak [39].…”

Section: Introductionmentioning

confidence: 91%

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A Shock-Capturing Algorithm for the Differential Equations of Dilation and Erosion

Breuß

Weickert

2006

J Math Imaging Vis

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Dilation and erosion are the fundamental operations in morphological image processing. Algorithms that exploit the formulation of these processes in terms of partial differential equations offer advantages for non-digitally scalable structuring elements and allow sub-pixel accuracy. However, the widely-used schemes from the literature suffer from significant blurring at discontinuities. We address this problem by developing a novel, flux corrected transport (FCT) type algorithm for morphological dilation / erosion with a flat disc. It uses the viscosity form of an upwind scheme in order to quantify the undesired diffusive effects. In a subsequent corrector step we compensate for these artifacts by means of a stabilised inverse diffusion process that requires a specific nonlinear multidimensional formulation. We prove a discrete maximum-minimum principle in this multidimensional framework. Our experiments show that the method gives a very sharp resolution of moving fronts, and it approximates rotation invariance very well.

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Section: Introductionmentioning

confidence: 91%

“…In some newer works mainly concerned with finite element schemes, antidiffusive fluxes are computed by algebraic properties of the entries of corresponding stiffness matrices, see e.g. [18] and the references therein. We employ a different approach motivated by the theory of numerical methods for conservation laws, compare e.g.…”

Section: Introductionmentioning

confidence: 99%

A Shock-Capturing Algorithm for the Differential Equations of Dilation and Erosion

Breuß

Weickert

2006

J Math Imaging Vis

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“…The new error indicators were applied to algebraic flux correction [14][15][16][17][18][19] schemes which were successfully equipped with grid adaptivity. The highly unstructured meshes resulting from local mesh refinement/coarsening call for fully implicit discretizations which are unconditionally stable.…”

Section: Discussionmentioning

confidence: 99%

“…The bisection process continues for all adjacent triangles sharing a hanging node with the refined element until all irregular grid points have been removed from the mesh. However, longest-edge bisection is mainly designed to uphold some geometric properties of the initial mesh and, thus, may not be the best comrade for our algebraic flux correction techniques [16][17][18][19] . For each element that needs to be refined due to accuracy reasons, the propagation path solely depends on the mesh geometry and does not account for the local solution behavior.…”

Section: Limited Gradient Reconstructionmentioning

confidence: 99%

“…In a series of recent publications [14][15][16][17][18][19] , an algebraic framework for the construction of high-resolution schemes for convection dominated partial differential equations was developed. The algebraic flux correction (AFC) paradigm renders a high-order discretization local extremum diminishing (LED) by a conservative elimination of negative off-diagonal entries from the discrete transport operator so as to end up with a non-oscillatory approximation of low order.…”

Section: Introductionmentioning

confidence: 99%

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On the Use of Slope Limiters for the Design of Recovery Based Error Indicators

Möller

Kuzmin

Numerical Mathematics and Advanced Applications

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Abstract. Gradient recovery techniques for the design of a posteriori error indicators are reviewed in the context of fluid dynamic problems featuring shocks and discontinuities. An edgewise slope limiting approach 24 tailored to linear finite element discretizations is presented. The improved gradient values at edge midpoints are recovered as the limited average of constant slopes from adjacent triangles. Furthermore, the low-order gradients may serve as natural upper and lower bounds to be imposed on the edge slopes. To this end, approved techniques such as averaging projection 33 , the (superconvergent) Zienkiewicz-Zhu patch recovery (SPR 34 ) and polynomial preserving recovery (PPR 29 ) are used to predict high-order gradients. A slope limiter is applied edge-by-edge to correct the provisional edge gradient values subject to geometric constraints. In either case, a second order accurate quadrature rule is employed to measure the difference between consistent and reconstructed slopes in the (local) L 2 -norm which provides a usable indicator for grid adaptation.The algebraic flux correction (AFC) methodology 14-19 is equipped with adaptive mesh refinement/coarsening procedures governed by the recovery based error indicator. The adaptive algorithm is applied to inviscid compressible flows at high Mach numbers.

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Simulation of Hot Isostatic Pressing in a Single‐Crystal Ni Base Superalloy with the Theory of Continuously Distributed Dislocations Combined with Vacancy Diffusion

2022

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Single‐crystal components made of nickel base superalloys contain pores after casting and homogenization heat treatment. Hot isostatic pressing (HIP), which is carried above the γ′‐solvus temperature of the alloy, is industrially applied to reduce porosity. A modeling of HIP based on continuously distributed dislocations is developed in a 2D setting. Glide and climb of straight‐edge dislocations, as well as vacancy diffusion, are the deformation mechanisms taken into account. Thereby, dislocation glide is controlled by dragging a cloud of large atoms, and climb is controlled by vacancy diffusion. Relying on previous investigations of the creep behavior at HIP temperatures, it is assumed that new dislocations are nucleated at low‐angle boundaries (LAB) and move through subgrains until they either reach the opposite LABs or react with other dislocations and annihilate. Vacancies are created at the pore surface and diffuse through the alloy until they are either consumed by climbing dislocations or disappear at the LABs. The field equations are solved by finite elements. It is shown that pore shrinking is mostly controlled by vacancy diffusion as the shear stresses at the LABs are too low to nucleate a sufficient amount of dislocations.

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Flux Correction Tools for Finite Elements

Cited by 171 publications

References 27 publications

A Shock-Capturing Algorithm for the Differential Equations of Dilation and Erosion

A Shock-Capturing Algorithm for the Differential Equations of Dilation and Erosion

On the Use of Slope Limiters for the Design of Recovery Based Error Indicators

Simulation of Hot Isostatic Pressing in a Single‐Crystal Ni Base Superalloy with the Theory of Continuously Distributed Dislocations Combined with Vacancy Diffusion

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