Efficient Solution Processes for Finite Element Analysis of Transient Heat Conduction

Gallagher, Richard H.; Mallett, R. H.

doi:10.1115/1.3449809

Cited by 25 publications

(4 citation statements)

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“…The mode superposition methods were originally proposed for transient heat diffusion problems [Biot, 1957;Gallagher and Mallet, 1971;Shih and Skladany, 1983], structural dynamics [Clough and Penzien, 1975], and the diffusion-convection equation [Nickell et al, 1979], and make use of a reduced transformation matrix which contains a number of the leftmost eigenvectors of a generalized eigenproblem with the same size as the finite element grid. Several techniques exist for finding the eigenpairs of sparse matrices [e.g., Parlett, 1980;Bathe, 1982;Hughes, 1987] and computer routines are also available [Bathe and Wilson, Paper number 92WR02331.…”

Section: Introductionmentioning

confidence: 99%

On time integration of groundwater flow equations by spectral methods

Gambolati¹

1993

Water Resources Research

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A study of the computational performance of the recently introduced spectral approach for the time integration of finite element flow equations is performed. Two reduction methods are presented and discussed. One method relies on the classical modal decomposition wherein the leftmost eigenpairs of a generalized symmetric eigenvalue problem are evaluated one at a time by a cost-effective orthogonal deflation technique. The other uses a set of Lanczos vectors for the coordinate transformation matrix and involves a recursive product between the inverse of the stiffness matrix and a vector. An extensive analysis of the behavior of the two methods is made with a representative sample problem of subsurface flow solved over a regular and an irregular mesh. The results show that the spectral approach cannot be competitive from a computational viewpoint with a more traditional time-stepping or direct integration scheme (e.g., the Crank-Nicolson scheme). The eigenvector technique is slow to converge, and a large portion, if not all, of the "flow modes" need to be determined to accurately describe the desired solution. The Lanczos algorithm may converge faster in the Lanczos vector space. However, its actual performance depends to a large extent on the initial guessed vector and is adversely influenced by the requirement to solve an equivalent steady state problem at each iteration. It is concluded that for a robust and efficient finite element code of transient groundwater flow, direct integration methods are generally to be preferred.

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Section: Introductionmentioning

confidence: 99%

On time integration of groundwater flow equations by spectral methods

Gambolati¹

1993

Water Resources Research

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“…Due to the appearance of the K~~, ~1 3 , ~2 3 , elements of the conductivity tensor, equation (5) has a non-canonical form.6.12 Hence as has been pointed out by Padovan for conduction problems in anisotropic thin walled shells of revolution,6 the traditional usage of Fourier series fails to yield a solution to equations (947). Furthermore, any attempts to reduce equations ( 3 4 7 ) to one of the canonical forms, renders the boundary value problem analytically intractable.…”

Section: Complex Series Representationmentioning

confidence: 99%

Semi‐analytical finite element procedure for conduction in anisotropic axisymmetric solids

Padovan

1974

Numerical Meth Engineering

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SUMMARYThrough the use of complex series representations and the properties of adjoint differential operators, a semi-analytical finite element procedure is developed which can analyse the steady state and transient temperature fields of thermally anisotropic axisymmetric bodies with complete circumferential properties. Due to its generality, the procedure can handle arbitrary laminate construction with possible meridional and radial variations in locally or globally thermally anisotropic materials. In this respect, the procedure developed herein supercedes the classical semi-analytical treatment which is limited to specially orthotropic materials. Furthermore, in contrast with the classical procedure, the present treatment reveals several important effects of material anisotropy which are entirely missed by the specially orthotropic assumption.

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“…Nevertheless, in the area of conductive heat transfer, the finite difference method is still a widely used method. However, the finite element method has also been used to solve steady state [17,18] and transient heat conduction problems [19,20] and it seems to gain popularity. Enery and Carson [21] have compared the FDM and FEM and concluded that the two methods are comparable for some constant-property steady state solutions but the FDM is better for most other cases.…”

Section: Introductionmentioning

confidence: 99%

Error Estimates for the Finite Difference Solution of the Heat Conduction Equation: Consideration of Boundary Conditions and Heat Sources

Davoudi

Öchsner²

2013

DDF

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This contribution investigates the numerical solution of the steady-state heat conduction equation. The finite difference method is applied to simple formulations of heat sources where still analytical solutions can be derived. Thus, the results of the numerical approach can be related to the exact solutions and conclusions on the accuracy obtained. In addition, the numerical implementation of different forms of boundary conditions, i.e. temperature and flux condition, is compared to the exact solution. It is found that the numerical implementation of coordinate dependent sources gives the exact result while temperature dependent sources are only approximately represented. Furthermore, the implementation of the mentioned boundary conditions gives the same results as the analytical reference solution.

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Efficient Solution Processes for Finite Element Analysis of Transient Heat Conduction

Cited by 25 publications

References 0 publications

On time integration of groundwater flow equations by spectral methods

On time integration of groundwater flow equations by spectral methods

Semi‐analytical finite element procedure for conduction in anisotropic axisymmetric solids

Error Estimates for the Finite Difference Solution of the Heat Conduction Equation: Consideration of Boundary Conditions and Heat Sources

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