Adaptive isogeometric finite element analysis of steady-state groundwater flow
Summary Numerical challenges occur in the simulation of groundwater flow problems because of complex boundary conditions, varying material properties, presence of sources or sinks in the flow domain, or a combination of these. In this paper, we apply adaptive isogeometric finite element analysis usi...
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| Published in | International journal for numerical and analytical methods in geomechanics Vol. 40; no. 5; pp. 738 - 765 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Bognor Regis
Blackwell Publishing Ltd
10.04.2016
Wiley Subscription Services, Inc |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0363-9061 1096-9853 |
| DOI | 10.1002/nag.2425 |
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| Abstract | Summary
Numerical challenges occur in the simulation of groundwater flow problems because of complex boundary conditions, varying material properties, presence of sources or sinks in the flow domain, or a combination of these. In this paper, we apply adaptive isogeometric finite element analysis using locally refined (LR) B‐splines to address these types of problems. The fundamentals behind isogeometric analysis and LR B‐splines are briefly presented. Galerkin's method is applied to the standard weak formulation of the governing equation to derive the linear system of equations. A posteriori error estimates are calculated to identify which B‐splines should be locally refined. The error estimates are calculated based on recovery of the L2‐projected solution. The adaptive analysis method is first illustrated by performing simulation of benchmark problems with analytical solutions. Numerical applications to two‐dimensional groundwater flow problems are then presented. The problems studied are flow around an impervious corner, flow around a cutoff wall, and flow in a heterogeneous medium. The convergence rates obtained with adaptive analysis using local refinement were, in general, observed to be of optimal order in contrast to simulations with uniform refinement. Copyright © 2015 John Wiley & Sons, Ltd. |
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| AbstractList | Numerical challenges occur in the simulation of groundwater flow problems because of complex boundary conditions, varying material properties, presence of sources or sinks in the flow domain, or a combination of these. In this paper, we apply adaptive isogeometric finite element analysis using locally refined (LR) B-splines to address these types of problems. The fundamentals behind isogeometric analysis and LR B-splines are briefly presented. Galerkin's method is applied to the standard weak formulation of the governing equation to derive the linear system of equations. A posteriori error estimates are calculated to identify which B-splines should be locally refined. The error estimates are calculated based on recovery of the L sub(2)-projected solution. The adaptive analysis method is first illustrated by performing simulation of benchmark problems with analytical solutions. Numerical applications to two-dimensional groundwater flow problems are then presented. The problems studied are flow around an impervious corner, flow around a cutoff wall, and flow in a heterogeneous medium. The convergence rates obtained with adaptive analysis using local refinement were, in general, observed to be of optimal order in contrast to simulations with uniform refinement. Summary Numerical challenges occur in the simulation of groundwater flow problems because of complex boundary conditions, varying material properties, presence of sources or sinks in the flow domain, or a combination of these. In this paper, we apply adaptive isogeometric finite element analysis using locally refined (LR) B‐splines to address these types of problems. The fundamentals behind isogeometric analysis and LR B‐splines are briefly presented. Galerkin's method is applied to the standard weak formulation of the governing equation to derive the linear system of equations. A posteriori error estimates are calculated to identify which B‐splines should be locally refined. The error estimates are calculated based on recovery of the L2‐projected solution. The adaptive analysis method is first illustrated by performing simulation of benchmark problems with analytical solutions. Numerical applications to two‐dimensional groundwater flow problems are then presented. The problems studied are flow around an impervious corner, flow around a cutoff wall, and flow in a heterogeneous medium. The convergence rates obtained with adaptive analysis using local refinement were, in general, observed to be of optimal order in contrast to simulations with uniform refinement. Copyright © 2015 John Wiley & Sons, Ltd. Numerical challenges occur in the simulation of groundwater flow problems because of complex boundary conditions, varying material properties, presence of sources or sinks in the flow domain, or a combination of these. In this paper, we apply adaptive isogeometric finite element analysis using locally refined (LR) B‐splines to address these types of problems. The fundamentals behind isogeometric analysis and LR B‐splines are briefly presented. Galerkin's method is applied to the standard weak formulation of the governing equation to derive the linear system of equations. A posteriori error estimates are calculated to identify which B‐splines should be locally refined. The error estimates are calculated based on recovery of the L 2 ‐projected solution. The adaptive analysis method is first illustrated by performing simulation of benchmark problems with analytical solutions. Numerical applications to two‐dimensional groundwater flow problems are then presented. The problems studied are flow around an impervious corner, flow around a cutoff wall, and flow in a heterogeneous medium. The convergence rates obtained with adaptive analysis using local refinement were, in general, observed to be of optimal order in contrast to simulations with uniform refinement. Copyright © 2015 John Wiley & Sons, Ltd. Summary Numerical challenges occur in the simulation of groundwater flow problems because of complex boundary conditions, varying material properties, presence of sources or sinks in the flow domain, or a combination of these. In this paper, we apply adaptive isogeometric finite element analysis using locally refined (LR) B-splines to address these types of problems. The fundamentals behind isogeometric analysis and LR B-splines are briefly presented. Galerkin's method is applied to the standard weak formulation of the governing equation to derive the linear system of equations. A posteriori error estimates are calculated to identify which B-splines should be locally refined. The error estimates are calculated based on recovery of the L2-projected solution. The adaptive analysis method is first illustrated by performing simulation of benchmark problems with analytical solutions. Numerical applications to two-dimensional groundwater flow problems are then presented. The problems studied are flow around an impervious corner, flow around a cutoff wall, and flow in a heterogeneous medium. The convergence rates obtained with adaptive analysis using local refinement were, in general, observed to be of optimal order in contrast to simulations with uniform refinement. Copyright © 2015 John Wiley & Sons, Ltd. |
| Author | Kvarving, Arne M. Bekele, Yared W. Nordal, Steinar Kvamsdal, Trond |
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| References_xml | – reference: Finn WDL. Finite-element analysis of seepage through dams. Journal of Soil Mechanics & Foundations Div, ASCE SM6. 1967; 92:41-48. – reference: Yeh GT. On the computation of Darcian velocity and mass balance in the finite-element modeling of groundwater-flow. Water Resources Research. 1981; 17(5):1529-1534. – reference: Zienkiewicz OC, Zhu JZ. The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique. International Journal for Numerical Methods in Engineering. 1992; 33(7):1331-1364. – reference: Jie YX, Liu LZ, Xu WJ, Li GX. Application of NEM in seepage analysis with a free surface. Mathematics and Computers in Simulation. 2013; 89:23-37. – reference: Burkley VJ, Bruch JC. Adaptive error analysis in seepage problems. International Journal for Numerical Methods in Engineering. 1991; 31(7):1333-1356. – reference: Tharp TM. An enriched finite-element for simulation of groundwater-flow to a well or drain. 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Numerical challenges occur in the simulation of groundwater flow problems because of complex boundary conditions, varying material properties, presence... Numerical challenges occur in the simulation of groundwater flow problems because of complex boundary conditions, varying material properties, presence of... Summary Numerical challenges occur in the simulation of groundwater flow problems because of complex boundary conditions, varying material properties, presence... |
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| SubjectTerms | a posteriori error estimates adaptive refinement Boundary conditions Computer simulation Errors Estimates Finite element method Galerkin methods Ground-water flow Groundwater flow isogeometric analysis Mathematical analysis Mathematical models |
| Title | Adaptive isogeometric finite element analysis of steady-state groundwater flow |
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