Formulation of a new error indicator for adaptive meshing with Hermite boundary elements

Muci-Küchler, Karim H.; Miranda-Valenzuela, J.C.; Rudolphi, Thomas J.

doi:10.1016/s0045-7825(98)00295-3

Cited by 7 publications

(11 citation statements)

References 11 publications

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“…In those cases, Muci-Küchler et al, [27] and Muci-Küchler and Miranda-Valenzuela [19] have shown that when Hermite elements are used, it is possible to eliminate the need of performing two different BEM solutions at each step of the adaptive process. Since the Hermite elements involve in their formulation the nodal values of the field variables and their tangential derivatives, it is possible to obtain two different approximate solutions for the tangential derivatives from only one analysis with Hermite elements.…”

Section: Formulation Of the Error Indicatorsmentioning

confidence: 99%

“…[27] for more details). As a rule, always the difference will be measured in terms of the quantity that was obtained from the solution of the system of equations generated by the BEM.…”

Section: Heat Conduction Problemmentioning

confidence: 99%

“…Muci-Küchler et al [27] and Muci-Küchler and MirandaValenzuela [19] presented formulations for the computation of simple error indicators for Hermite elements in twodimensional potential and elastostatic problems. These error indicators were based on the idea of obtaining two different approximate numerical solutions for the tangential derivatives of the field variables from just one analysis with Hermite elements.…”

Section: Introductionmentioning

confidence: 99%

“…[19,27] are extended to perform adaptive meshing with Hermite elements in steady-state, uncoupled, two-dimensional thermoelastic problems. Since the thermoelastic problem in its uncoupled fashion is considered, it is possible to solve the heat conduction and thermoelastic problems separately, solving first for the temperature distribution on the boundary, and then using that information to solve for the boundary stresses.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, a more general approach than the one given in Ref. [27] for the computation of the error indicator for the potential problem is also presented, tackling some of the restrictions given in that reference. Numerical examples are included to illustrate the versatility of the formulation and its numerical implementation.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive meshing for two-dimensional thermoelastic problems using Hermite boundary elements

Miranda-Valenzuela

Muci-Küchler

Soriano-Soriano³

2001

Advances in Engineering Software

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In the present work, error indicators for the potential and elastostatic problems are used in a combined fashion to implement an adaptive meshing scheme for the solution of two-dimensional steady-state thermoelastic problems using the Boundary Element Method. These error indicators exploit in their formulation the possibility of generating two different numerical solutions from just one analysis using Hermite elements. The first solution is the standard one obtained from an analysis using Hermite elements. The second is a "reduced" solution obtained representing the field variables inside an element using some of the degrees of freedom of the Hermite element together with Lagrangian shape functions. The basic idea behind the computation of the error indicator is to compare these two solutions, on an element by element basis, to obtain an estimate of the magnitude of the error in the numerical solution corresponding to the Hermite elements. In this sense, it is assumed that the bigger the difference between these two solutions, the bigger the error in the original solution with Hermite elements. Since the thermoelastic problem in its uncoupled fashion is considered, the former approach is applied to both problems, heat conduction and thermoelastic. Since both numerical solutions for each one of these problems are obtained from just one analysis, the computational cost of the proposed error indicators is very low. ᭧

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Section: Formulation Of the Error Indicatorsmentioning

confidence: 99%

“…[27] for more details). As a rule, always the difference will be measured in terms of the quantity that was obtained from the solution of the system of equations generated by the BEM.…”

Section: Heat Conduction Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Adaptive meshing for two-dimensional thermoelastic problems using Hermite boundary elements

Miranda-Valenzuela

Muci-Küchler

Soriano-Soriano³

2001

Advances in Engineering Software

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Boundary integral equation for tangential derivative of flux in Laplace and Helmholtz equations

Gallego¹,

Martínez-Castro²

2006

Int. J. Numer. Meth. Engng

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SUMMARYIn this paper, the boundary integral equations (BIEs) for the tangential derivative of flux in Laplace and Helmholtz equations are presented. These integral representations can be used in order to solve several problems in the boundary element method (BEM): cubic solutions including degrees of freedom in flux's tangential derivative value (Hermitian interpolation), nodal sensitivity, analytic gradients in optimization problems, or tangential derivative evaluation in problems that require the computation of such variable (elasticity problems in BEM). The analysis has been developed for 2D formulation. Kernels for tangential derivative of flux lead to high-order singularities (O(1/r 3 )). The limit to the boundary analysis has been carried out. Based on this analysis, regularization formulae have been obtained in order to use such BIE in numerical codes. A set of numerical benchmarks have been carried out in order to validate theoretical and practical aspects, by considering known analytic solutions for the test problems. The results show that the tangential BIEs have been properly developed and implemented.

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Use of the tangent derivative boundary integral equations for the efficient computation of stresses and error indicators

Muci-Küchler

Miranda-Valenzuela

Soriano-Soriano³

2001

Numerical Meth Engineering

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SUMMARYIn this work, a new global reanalysis technique for the e cient computation of stresses and error indicators in two-dimensional elastostatic problems is presented. In the context of the boundary element method, the global reanalysis technique can be viewed as a post-processing activity that is carried out once an analysis using Lagrangian elements has been performed. To do the reanalysis, the functional representation for the displacements is changed from Lagrangian to Hermite, introducing the nodal values of the tangential derivatives of those quantities as additional degrees of freedom. Next, assuming that the nodal values of the displacements and the tractions remain practically unchanged from the ones obtained in the analysis using Lagrangian elements, the tangent derivative boundary integral equations are collocated at each functional node in order to determine the additional degrees of freedom that were introduced. Under this scheme, a second system of equations is generated and, once it is solved, the nodal values of the tangential derivatives of the displacements are obtained. This approach gives more accurate results for the stresses at the nodes since it avoids the need to di erentiate the shape functions in order to obtain the normal strain in the tangential direction. When compared with the use of Hermite elements, the global reanalysis technique has the attraction that the user does not have to give as input data the additional information required by this type of elements. Another important feature of the proposed approach is that an e cient error indicator for the values of the stresses can also be obtained comparing the values for the stresses obtained through the use of Lagrangian elements and the global reanalysis technique.

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Formulation of a new error indicator for adaptive meshing with Hermite boundary elements

Cited by 7 publications

References 11 publications

Adaptive meshing for two-dimensional thermoelastic problems using Hermite boundary elements

Adaptive meshing for two-dimensional thermoelastic problems using Hermite boundary elements

Boundary integral equation for tangential derivative of flux in Laplace and Helmholtz equations

Use of the tangent derivative boundary integral equations for the efficient computation of stresses and error indicators

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