scholarly journals Finite element solution of nonlocal Cahn‐Hilliard equations with feedback control time step size adaptivity

2021 ◽  
Author(s):  
Gabriel F. Barros ◽  
Adriano M. A. Côrtes ◽  
Alvaro L. G. A. Coutinho
Keyword(s):  
Finite Element ◽  
Feedback Control ◽  
Finite Element Solution ◽  
Element Solution ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Control Time

scholarly journals A global finite-element shallow-water model supporting continuous and discontinuous elements

2014 ◽  
Vol 7 (6) ◽  
pp. 3017-3035 ◽  
Author(s):  
P. A. Ullrich
Keyword(s):  
Finite Element ◽  
Shallow Water ◽  
Wave Train ◽  
Correction Function ◽  
Water Model ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Discontinuous Elements ◽  
Cubed Sphere

Abstract. This paper presents a novel nodal finite-element method for either continuous and discontinuous elements, as applied to the 2-D shallow-water equations on the cubed sphere. The cornerstone of this method is the construction of a robust derivative operator that can be applied to compute discrete derivatives even over a discontinuous function space. A key advantage of the robust derivative is that it can be applied to partial differential equations in either a conservative or a non-conservative form. However, it is also shown that discontinuous penalization is required to recover the correct order of accuracy for discontinuous elements. Two versions with discontinuous elements are examined, using either the g1 and g2 flux correction function for distribution of boundary fluxes and penalty across nodal points. Scalar and vector hyperviscosity (HV) operators valid for both continuous and discontinuous elements are also derived for stabilization and removal of grid-scale noise. This method is validated using four standard shallow-water test cases, including geostrophically balanced flow, a mountain-induced Rossby wave train, the Rossby–Haurwitz wave and a barotropic instability. The results show that although the discontinuous basis requires a smaller time step size than that required for continuous elements, the method exhibits better stability and accuracy properties in the absence of hyperviscosity.

Finite-Element Method Applied to Heat Conduction in Solids with Nonlinear Boundary Conditions

1973 ◽  
Vol 95 (1) ◽  
pp. 126-129 ◽  
Author(s):  
R. E. Beckett ◽  
S.-C. Chu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Transient Heat Conduction ◽  
Nonlinear Boundary Conditions ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Element Method

By use of an implicit iteration technique, the finite-element method applied to the heat-conduction problems of solids is no longer restricted to the linear heat-flux boundary conditions, but is extended to include nonlinear radiation–convection boundary conditions. The variation of surface temperatures within each time increment is taken into account; hence a rather large time-step size can be assigned to obtain transient heat-conduction solutions without introducing instability in the surface temperature of a body.

scholarly journals Comparison between numerical analysis and the levitation mass method measurement test of a spherical structure early impacting water

2018 ◽  
Vol 10 (1) ◽  
pp. 168781401774807 ◽  
Author(s):  
Yonghu Wang ◽  
Dongwei Shu ◽  
Yusaku Fujii ◽  
Akihiro Takita ◽  
Tsuneaki Ishima ◽  
...  
Keyword(s):  
Finite Element ◽  
Contact Stiffness ◽  
Experimental Results ◽  
Impact Loads ◽  
Arbitrary Lagrangian Eulerian ◽  
Water Impact ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Spherical Structure

In order to precisely measure water impact loads of a spherical structure vertically dropping onto a calm water surface, a new validity check of the analysis using the levitation mass method experiment is proposed. The main feature of levitation mass method experiment is to obtain a better estimation of early water impact loads through the application of Doppler effect. Experimental results of different heights are verified based on the Assessment Index and are in comparison with the classical experimental data for validation purpose. It shows that the levitation mass method measurement is useful and effective to obtain the water impact loads for the crashworthiness analysis. Besides, early water impact hydrodynamic behaviors are simulated based on the nonlinear explicit finite element method, together with application of a multi-material arbitrary Lagrangian–Eulerian solver. A penalty coupling algorithm is utilized to realize fluid–structure interaction between the spherical body and fluids. Convergence studies are performed to construct the proper finite element model by the comparison with experimental results, where mesh sensitivity, contact stiffness, and time-step size parametric studies are thoroughly investigated. The comparisons between experimental and numerical results show good consistency by the prediction of the water impact coefficients on the structure.

The Dispersion in One- and Two-Dimensional Finite Element Solutions to the Heat Equation

2002 ◽  
Author(s):  
W. Dauksher ◽  
A. F. Emery
Keyword(s):  
Finite Element ◽  
Heat Equations ◽  
Two Dimensional ◽  
Step Size ◽  
Matrix Formulation ◽  
Time Step ◽  
Time Step Size ◽  
Time Stepping ◽  
Dimensional Finite Element ◽  
The One

The dispersive errors in the finite element solution to the one- and two-dimensional heat equations are examined as a function of element type and size, capacitance matrix formulation, time stepping scheme and time step size.

Numerical oscillation in seepage analysis of unsaturated soils

2001 ◽  
Vol 38 (3) ◽  
pp. 639-651 ◽  
Author(s):  
Muthusamy Karthikeyan ◽  
Thiam-Soon Tan ◽  
Kok-Kwang Phoon
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Unsaturated Soil ◽  
Unsaturated Soils ◽  
Heat Diffusion ◽  
Step Size ◽  
Time Step ◽  
Seepage Analysis ◽  
Time Step Size ◽  
Element Method

The finite element method provides a popular means of analyzing groundwater flow in an unsaturated soil. In such problems, oscillatory results are often observed in the finite element solution. Such a phenomenon is observed, for example, when a typical finite element program such as Seep/w is used to model water infiltration into unsaturated soils. Numerical oscillations are often found near the wetting front where the hydraulic gradient is the steepest. These oscillations do not always reduce with decreasing or increasing time-step size alone; rather, an appropriate ratio between time-step size and element size is required. As the pore-water pressures predicted from a transient seepage analysis are used as input groundwater conditions for other types of analysis such as slope stability, contaminant transport, and capillary barrier, these oscillations may have important practical ramifications. Since seepage analysis is common in engineering practice, it is important that appropriate criteria are identified to minimize, if not to remove, the oscillations. In this paper, numerical examples are provided to demonstrate that a simple set of criteria, developed in heat diffusion problems with constant properties to control oscillation, is also applicable to one- and two-dimensional unsaturated seepage analyses, for a range of material nonlinearities that are frequently encountered in unsaturated soils.Key words: unsaturated soil, soil-water characteristic curve, seepage analysis, finite element method, numerical oscillation.

scholarly journals Energy-corrected FEM and explicit time-stepping for parabolic problems

2019 ◽  
Vol 53 (6) ◽  
pp. 1893-1914
Author(s):  
Piotr Swierczynski ◽  
Barbara Wohlmuth
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Regularity Of Solutions ◽  
Parabolic Problems ◽  
Element Approximation ◽  
Computational Domain ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Stepping

The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called “pollution effect”. Standard remedies based on mesh refinement around the singular corner result in very restrictive stability requirements on the time-step size when explicit time integration is applied. In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation. We illustrate these results with extensive numerical investigations not only confirming the theoretical results but also showing the flexibility of the method, which can be applied in the presence of multiple singular corners and a three-dimensional setting. We also propose a fast explicit time-stepping scheme based on a piecewise cubic energy-corrected discretisation in space completed with mass-lumping techniques and numerically verify its efficiency.

Accurate Prediction of Strains and Stresses in 2-D Elasticity Using Adini’s Finite Element

2015 ◽  
Author(s):  
Donovan A. Aguirre-Rivas ◽  
Karim H. Muci-Küchler
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Computational Cost ◽  
Finite Element Solution ◽  
Element Solution ◽  
Time Step ◽  
Solution Approach ◽  
Linear Elastic ◽  
Element Size ◽  
Linearized Problem

In the interest of obtaining accurate stress predictions in linear elastic problems while keeping the computational cost low, a finite element solution approach using cubic elements that include not only the displacement but also the spatial derivatives of the displacement as nodal degrees of freedom (DOFs) is explored in this paper. The proposed approach has the advantage that the nodal values of the strains, and hence the stresses, can be directly computed from the finite element solution and, as shown in this paper, it is capable of converging faster to the analytical solution than the commonly used reduced integration Serendipity quadratic element. Because the proposed approach is capable of achieving high accuracy using less DOFs, it is possible to use coarser meshes than with conventional elements. This is of particular importance in dynamic problems in which explicit techniques are used and the size of the time step is tied to the element size. Moreover, the proposed approach can be beneficial in non-linear problems in which stepping techniques are used to solve a linearized problem and the strains or stresses of the current step are used as input for the following step.

scholarly journals A global finite-element shallow-water model supporting continuous and discontinuous elements

2014 ◽  
Vol 7 (4) ◽  
pp. 5141-5182 ◽  
Author(s):  
P. A. Ullrich
Keyword(s):  
Finite Element ◽  
Shallow Water ◽  
Wave Train ◽  
Correction Function ◽  
Water Model ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Discontinuous Elements ◽  
Cubed Sphere

Abstract. This paper presents a novel nodal finite element method for either continuous and discontinuous elements, as applied to the 2-D shallow-water equations on the cubed-sphere. The cornerstone of this method is the construction of a robust derivative operator which can be applied to compute discrete derivatives even over a discontinuous function space. A key advantage of the robust derivative is that it can be applied to partial differential equations in either conservative or non-conservative form. However, it is also shown that discontinuous penalization is required to recover the correct order of accuracy for discontinuous elements. Two versions with discontinuous elements are examined, using either the g1 and g2 flux correction function for distribution of boundary fluxes and penalty across nodal points. Scalar and vector hyperviscosity operators valid for both continuous and discontinuous elements are also derived for stabilization and removal of grid-scale noise. This method is validated using three standard shallow-water test cases, including geostrophically balanced flow, a mountain-induced Rossby wave train and a barotropic instability. The results show that although the discontinuous basis requires a smaller time step size than that required for continuous elements, the method exhibits better stability and accuracy properties in the absence of hyperviscosity.

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