A comparison of domain integral evaluation techniques for boundary element methods

Ingber, M. S.; Mammoli, Andrea; Brown, Mary J.

doi:10.1002/nme.217

Cited by 56 publications

(42 citation statements)

References 13 publications

(7 reference statements)

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“…In order to extend the work performed by Ingber et al [5], the same potential problem is used, namely…”

Section: Results and Analysismentioning

confidence: 99%

“…The augmented function at all integration points only needs to be evaluated once, so the linear combination of linear basis functions (Equation (6)) does not require appreciable additional computation. The evaluation of the integral over the augmented computational domain, and on each auxiliary domain, can be performed by fast multipole methods, as was done by Ingber et al [5], with potentially orders of magnitude savings in computational e ort, particularly as the problem size increases.…”

Section: Discussionmentioning

confidence: 99%

“…Second, with recent developments in the rapid evaluation of integrals, namely the application of fast multipole methods (FMM), solution of BEM problems with direct domain integral evaluation may be considerably faster than with meshless methods. In fact, a recent study has shown that cell-based methods are simultaneously orders of magnitude faster and more accurate than meshless methods [5]. Third, domain discretization does not lead to an increase in the size of the linear system, in contrast to meshless methods.…”

Section: Introductionmentioning

confidence: 98%

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Solution of non‐linear boundary integral equations in complex geometries with auxiliary integral subtraction

Mammoli¹

2002

Numerical Meth Engineering

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SUMMARYThe boundary integral equation that results from the application of the reciprocity theorem to nonlinear or non-homogeneous di erential equations generally contains a domain integral. While methods exist for the meshless evaluation of these integrals, mesh-based domain integration is generally more accurate and can be performed more quickly with the application of fast multipole methods. However, polygonalization of complex multiply-connected geometries can become a costly task, especially in three-dimensional analyses. In this paper, a method that allows a mesh-based integration in complex domains, while retaining a simple mesh structure, is described. Although the technique is intended for the numerical solution of more complex di erential equations, such as the Navier-Stokes equations, for simplicity the method is applied to the solution of a Poisson equation, in domains of varying complexity. It is shown that the error introduced by the auxiliary domain subtraction method is comparable to the discretization error, while the complexity of the mesh is signiÿcantly reduced. The behaviour of the error in the boundary solution observed with the application of the new method is analogous to the behaviour observed with conventional cell-based domain integration.

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“…In order to extend the work performed by Ingber et al [5], the same potential problem is used, namely…”

Section: Results and Analysismentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Solution of non‐linear boundary integral equations in complex geometries with auxiliary integral subtraction

Mammoli¹

2002

Numerical Meth Engineering

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“…Of the many methods for dealing with this, the direct cell-based evaluation of the integral appears to be the most accurate, as demonstrated by Ingber et al (2001). The multipole algorithm can be applied to the domain integration, resulting in significant reductions in calculation times.…”

Section: Bem Codementioning

confidence: 99%

Maximizing Storage Rate and Capacity and Insuring the Environmental Integrity of Carbon Dioxide Sequestration in Geological Reservoirs

Davis¹,

Graham²,

Parker³

et al. 2005

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Maximizing Storage Rate and Capacity and Insuring the Environmental Integrity of Carbon Dioxide Sequestration in Geological FormationsThe U.S. and other countries may enter into an agreement that will require a significant reduction in CO 2 emissions in the medium to long term. In order to achieve such goals without drastic reductions in fossil fuel usage, CO 2 must be removed from the atmosphere and be stored in acceptable reservoirs. The research outlined in this proposal deals with developing a methodology to determine the suitability of a particular geologic formation for the long-term storage of CO 2 and technologies for the economical transfer and storage of CO 2 in these formations. A novel well-logging technique using nuclear-magnetic resonance (NMR) will be developed to characterize the geologic formation including the integrity and quality of the reservoir seal (cap rock). Well-logging using NMR does not require coring, and hence, can be performed much more quickly and efficiently. The key element in the economical transfer and storage of the CO 2 is hydraulic fracturing the formation to achieve greater lateral spreads and higher throughputs of CO 2 . Transport, compression, and drilling represent the main costs in CO 2 sequestration. The combination of well-logging and hydraulic fracturing has the potential of minimizing these costs. It is possible through hydraulic fracturing to reduce the number of injection wells by an order of magnitude.Many issues will be addressed as part of the proposed research to maximize the storage rate and capacity and insure the environmental integrity of CO 2 sequestration in geological formations. First, correlations between formation properties and NMR relaxation times will be firmly established. A detailed experimental program will be conducted to determine these correlations. Second, improved hydraulic fracturing models will be developed which are suitable for CO 2 sequestration as opposed to enhanced oil recovery (EOR). Although models that simulate the fracturing process exist, they can be significantly improved by extending the models to account for nonsymmetric, nonplanar fractures, coupling the models to more realistic reservoir simulators, and implementing advanced multiphase flow models for the transport of proppant. Third, it may be possible to deviate from current hydraulic fracturing technology by using different proppants (possibly waste materials that need to be disposed of, e.g., asbestos) combined with different hydraulic fracturing carrier fluids (possibly supercritical CO 2 itself). Because current technology is mainly aimed at enhanced oil recovery, it may not be ideally suited for the injection and storage of CO 2 . Finally, advanced concepts such as increasing the injectivity of the fractured geologic formations through acidization with carbonated water will be investigated. Saline formations are located through most of the continental United States. Generally, where saline formations are scarce, oil and gas reservoirs and coal beds abound...

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“…Although the neural network approach can also be used in the treatment of the volume integral via the particular solution technique (Nguyen-Thien and Tran-Cong [23]), it is not employed here since the aim is to show that neural network approximation can be a better alternative to the traditional boundary element approximation and this comparison can only be done if all other parameters are kept the same as those reported in the literature. Furthermore, there is evidence to show that neither accuracy (Ingber et al [24]) nor convergence (Power and Mingo [19]) is improved when volume integrals are approximated by similar techniques, such as dual reciprocity and particular solution. The results obtained by the present method such as the velocity proÿles along the horizontal and vertical centrelines as well as the properties of the primary vortex are in very good agreement with the benchmark solution.…”

Section: Introductionmentioning

confidence: 97%

Neural networks for BEM analysis of steady viscous flows

Mai‐Duy

Tran-Cong

2003

Numerical Methods in Fluids

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SUMMARYThis paper presents a new neural network-boundary integral approach for analysis of steady viscous uid ows. Indirect radial basis function networks (IRBFNs) which perform better than element-based methods for function interpolation, are introduced into the BEM scheme to represent the variations of velocity and traction along the boundary from the nodal values. In order to assess the e ect of IRBFNs, the other features used in the present work remain the same as those used in the standard BEM. For example, Picard-type scheme is utilized in the iterative procedure to deal with the non-linear convective terms while the calculation of volume integrals and velocity gradients are based on the linear ÿnite element-based method. The proposed IRBFN-BEM is veriÿed on the driven cavity viscous ow problem and can achieve a moderate Reynolds number of 1400 using a relatively coarse uniform mesh. The results obtained such as the velocity proÿles along the horizontal and vertical centrelines as well as the properties of the primary vortex are in very good agreement with the benchmark solution. Furthermore, the secondary vortices are also captured by the present method. Thus, it appears that an ability to represent the boundary solution accurately can signiÿcantly improve the overall solution accuracy of the BEM.

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A comparison of domain integral evaluation techniques for boundary element methods

Cited by 56 publications

References 13 publications

Solution of non‐linear boundary integral equations in complex geometries with auxiliary integral subtraction

Solution of non‐linear boundary integral equations in complex geometries with auxiliary integral subtraction

Maximizing Storage Rate and Capacity and Insuring the Environmental Integrity of Carbon Dioxide Sequestration in Geological Reservoirs

Neural networks for BEM analysis of steady viscous flows

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