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.

scholarly journals Numerical Gradient Schemes for Heat Equations Based on the Collocation Polynomial and Hermite Interpolation

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 93 ◽  
Author(s):  
Hou-Biao Li ◽  
Ming-Yan Song ◽  
Er-Jie Zhong ◽  
Xian-Ming Gu
Keyword(s):  
Hermite Interpolation ◽  
Mesh Size ◽  
Dirichlet Boundary Conditions ◽  
Maximum Norm ◽  
Heat Equations ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
The One ◽  
Delay Partial Differential Equations

As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is that it is unconditionally stable and convergent on the order O ( τ 2 + h 4 ) (where τ is the time step size and h is the mesh size), under the maximum norm for a class of nonlinear delay partial differential equations with initial and Dirichlet boundary conditions. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. The convergence order of this kind of method is also O ( τ 2 + h 4 ) under the discrete maximum norm when the spatial step size is twice the one of H-OCD, which accelerates the computational process. In addition, some corresponding analyses are made and the Richardson extrapolation technique is also considered in the time direction. The results of numerical experiments are consistent with the theoretical analysis.

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.

scholarly journals Analysis of a Decoupled Time-Stepping Scheme for Evolutionary Micropolar Fluid Flows

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
S. S. Ravindran
Keyword(s):  
Micropolar Fluid ◽  
Stokes Equations ◽  
Fluid Model ◽  
Navier Stokes ◽  
Optimal Order ◽  
Step Size ◽  
Time Step ◽  
Order Error ◽  
Time Step Size ◽  
Time Stepping

Micropolar fluid model consists of Navier-Stokes equations and microrotational velocity equations describing the dynamics of flows in which microstructure of fluid is important. In this paper, we propose and analyze a decoupled time-stepping algorithm for the evolutionary micropolar flow. The proposed method requires solving only one uncoupled Navier-Stokes and one microrotation subphysics problem per time step. We derive optimal order error estimates in suitable norms without assuming any stability condition or time step size restriction.

Preliminary Ship Design Using One and Two-Dimensional Models

1999 ◽  
Vol 36 (02) ◽  
pp. 102-112
Author(s):  
Michael D. A. Mackney ◽  
Carl T. F. Ross
Keyword(s):  
Finite Element ◽  
Dimensional Analysis ◽  
Three Dimensional ◽  
Two Dimensional ◽  
One Dimensional ◽  
Dimensional Finite Element ◽  
Dimensional Models ◽  
Support Conditions ◽  
The One ◽  
Three Dimensional Finite Element

Computational studies of hull-superstructure interaction were carried out using one-, two-and three-dimensional finite element analyses. Simplification of the original three-dimensional cases to one- and two-dimensional ones was undertaken to reduce the data preparation and computer solution times in an extensive parametric study. Both the one- and two-dimensional models were evaluated from numerical and experimental studies of the three-dimensional arrangements of hull and superstructure. One-dimensional analysis used a simple beam finite element with appropriately changed sections properties at stations where superstructures existed. Two-dimensional analysis used a four node, first order quadrilateral, isoparametric plane elasticity finite element, with a corresponding increase in the grid domain where the superstructure existed. Changes in the thickness property reflected deck stiffness. This model was essentially a multi-flanged beam with the shear webs representing the hull and superstructure sides, and the flanges representing the decks One-dimensional models consistently and uniformly underestimated the three-dimensional behaviour, but were fast to create and run. Two-dimensional models were also consistent in their assessment, and considerably closer in predicting the actual behaviours. These models took longer to create than the one-dimensional, but ran in very much less time than the refined three-dimensional finite element models Parametric insights were accomplished quickly and effectively with the simplest model and processor, but two-dimensional analyses achieved closer absolute measure of the displacement behaviours. Although only static analysis with simple loading and support conditions were presented, it is believed that similar benefits would be found for other loadings and support conditions. Other engineering components and structures may benefit from similarly judged simplification using one- and two-dimensional models to reduce the time and cost of preliminary design.

scholarly journals Investigation of Numerical Conditions of Moving Particle Semi-implicit for Two-Dimensional Wedge Slamming

2022 ◽  
Author(s):  
Takahito Iida ◽  
Yudai Yokoyama
Keyword(s):  
Particle Size ◽  
Poor Performance ◽  
Verification And Validation ◽  
Two Dimensional ◽  
Step Size ◽  
Time Step ◽  
Optimum Time ◽  
Time Step Size ◽  
Tuning Parameters ◽  
Moving Particle

AbstractThe sensitivity of moving particle semi-implicit (MPS) simulations to numerical parameters is investigated in this study. Although the verification and validation (V&V) are important to ensure accurate numerical results, the MPS has poor performance in convergences with a time step size. Therefore, users of the MPS need to tune numerical parameters to fit results into benchmarks. However, such tuning parameters are not always valid for other simulations. We propose a practical numerical condition for the MPS simulation of a two-dimensional wedge slamming problem (i.e., an MPS-slamming condition). The MPS-slamming condition is represented by an MPS-slamming number, which provides the optimum time step size once the MPS-slamming number, slamming velocity, deadrise angle of the wedge, and particle size are decided. The simulation study shows that the MPS results can be characterized by the proposed MPS-slamming condition, and the use of the same MPS-slamming number provides a similar flow.

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.

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