Are fem solutions of incompressible flows really incompressible? (or how simple flows can cause headaches!)

Pelletier, Dominique; Fortin, A.; Camarero, R.

doi:10.1002/fld.1650090108

Cited by 104 publications

(52 citation statements)

References 5 publications

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“…Hence, the P1±isoP2 element is weakly compressible. Pelletier et al (1989) emphasise one consequence of this behaviour with u and p the solutions of the continuous Stokes problem, v h and p h the solutions of the discrete problem. The accuracy of the velocity depends both on the approximations of the velocity and pressure.…”

Section: Spatial Discretizationmentioning

confidence: 98%

LES and RANS of turbulent flow in tube bundles

Rollet-Miet

Laurence

Ferziger

1999

International Journal of Heat and Fluid Flow

110

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Section: Spatial Discretizationmentioning

confidence: 98%

LES and RANS of turbulent flow in tube bundles

Rollet-Miet

Laurence

Ferziger

1999

International Journal of Heat and Fluid Flow

110

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“…Note that we see none of the pressure oscillation problems that are often caused by the incompressibility constraint [e.g., Pelletier et al, 1989].…”

Section: Comparison To Other Softwarementioning

confidence: 99%

MILAMIN: MATLAB‐based finite element method solver for large problems

Dąbrowski

Krotkiewski

Schmid

2008

Geochem Geophys Geosyst

228

145

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[1] The finite element method (FEM) combined with unstructured meshes forms an elegant and versatile approach capable of dealing with the complexities of problems in Earth science. Practical applications often require high-resolution models that necessitate advanced computational strategies. We therefore developed ''Million a Minute'' (MILAMIN), an efficient MATLAB implementation of FEM that is capable of setting up, solving, and postprocessing two-dimensional problems with one million unknowns in one minute on a modern desktop computer. MILAMIN allows the user to achieve numerical resolutions that are necessary to resolve the heterogeneous nature of geological materials. In this paper we provide the technical knowledge required to develop such models without the need to buy a commercial FEM package, programming compiler-language code, or hiring a computer specialist. It has been our special aim that all the components of MILAMIN perform efficiently individually and as a package. While some of the components rely on readily available routines, we develop others from scratch and make sure that all of them work together efficiently. One of the main technical focuses of this paper is the optimization of the global matrix computations. The performance bottlenecks of the standard FEM algorithm are analyzed. An alternative approach is developed that sustains high performance for any system size. Applied optimizations eliminate Basic Linear Algebra Subprograms (BLAS) drawbacks when multiplying small matrices, reduce operation count and memory requirements when dealing with symmetric matrices, and increase data transfer efficiency by maximizing cache reuse. Applying loop interchange allows us to use BLAS on large matrices. In order to avoid unnecessary data transfers between RAM and CPU cache we introduce loop blocking. The optimization techniques are useful in many areas as demonstrated with our MILAMIN applications for thermal and incompressible flow (Stokes) problems. We use these to provide performance comparisons to other open source as well as commercial packages and find that MILAMIN is among the best performing solutions, in terms of both speed and memory usage. The corresponding MATLAB source code for the entire MILAMIN, including input generation, FEM solver, and postprocessing, is available from the authors (http://www.milamin.org) and can be downloaded as auxiliary material.

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“…The rheology is visco-elasto-plastic with Mohr-Coulomb plasticity. A mixed formulation is employed, with linear, discontinuous, shape-functions for pressure and quadratic shape functions for velocity to avoid spurious pressures (Pelletier et al 1989). Uzawa-type iterations are used to satisfy the incompressibility constraint (Cuvelier et al 1986).…”

Section: Slomomentioning

confidence: 99%

The numerical sandbox: comparison of model results for a shortening and an extension experiment

et al. 2006

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Abstract:We report results of a study comparing numerical models of sandbox-type experiments. Two experimental designs were examined: (1) A brittle shortening experiment in which a thrust wedge is built in material of alternating frictional strength; and (2) an extension experiment in which a weak, basal viscous layer affects normal fault localization and propagation in overlying brittle materials. Eight different numerical codes, both commercial and academic, were tested against each other. Our results show that: (1) The overall evolution of all numerical codes is broadly similar. (2) Shortening is accommodated by in-sequence forward propagation of thrusts. The surface slope of the thrust wedge is within the stable field predicted by critical taper theory. (3) Details of thrust spacing, dip angle and number of thrusts vary between different codes for the shortening experiment. (4) Shear zones initiate at the velocity discontinuity in the extension experiment. The asymmetric evolution of the models is similar for all numerical codes. (5) Resolution affects strain localization and the number of shear zones that develop in strain-softening brittle material. (6) The variability between numerical codes is greater for the shortening than the extension experiment.Comparison to equivalent analogue experiments shows that the overall dynamic evolution of the numerical and analogue models is similar, in spite of the difficulty of achieving an exact representation of the analogue conditions with a numerical model. We find that the degree of variability between individual numerical results is about the same as between individual analogue models. Differences among and between numerical and analogue results are found in predictions of location, spacing and dip angle of shear zones. Our results show that numerical models using different solution techniques can to first order successfully reproduce structures observed in analogue sandbox experiments. The comparisons serve to highlight robust features in tectonic modelling of thrust wedges and brittle-viscous extension.Numerical and analogue modelling methods represent two different techniques with which the evolution of geological structures, such as fold-and-thrust belts and sedimentary basins, can be investigated. The underlying assumption with both methods is that their results approximate the development of structures in the real Earth in a reasonable manner. We may then expect that the results of analogue and numerical models look similar when applied to the same (2) to test the similarity of numerical and analogue models, in order to help establish robust features of tectonic models on the scale of the upper crust.The companion paper (Schreurs et al. 2006) presents the results of an analogue comparison study with ten participating modelling laboratories. Two experimental set-ups were tested: (1) a brittle convergent thrust wedge experiment and (2) a brittle-viscous extension experiment. The reproducibility of modelling results between the laboratories was found to be fai...

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Are fem solutions of incompressible flows really incompressible? (or how simple flows can cause headaches!)

Cited by 104 publications

References 5 publications

LES and RANS of turbulent flow in tube bundles

LES and RANS of turbulent flow in tube bundles

MILAMIN: MATLAB‐based finite element method solver for large problems

The numerical sandbox: comparison of model results for a shortening and an extension experiment

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