Efficient simulation of unsaturated moisture flow in porous media is of great importance in many engineering fields. The highly non-linear character of unsaturated flow typically gives sharp moving moisture fronts during wetting and drying of materials with strong local moisture permeability and capacity variations as result. It is shown that these strong variations conflict with the common preference for low-order numerical integration in finite element simulations of unsaturated moisture flow: inaccurate numerical integration leads to errors that are often far more important than errors from inappropriate discretization.In response, this article develops adaptive integration, based on nested Kronrod-Patterson-Gauss integration schemes: basically, the integration order is adapted to the locally observed grade of non-linearity. Adaptive integration is developed based on a standard infiltration problem, and it is demonstrated that serious reductions in the numbers of required integration points and discretization nodes can be obtained, thus significantly increasing computational efficiency. The multi-dimensional applicability is exemplified with two-dimensional wetting and drying applications.While developed for finite element unsaturated moisture transfer simulation, adaptive integration is similarly applicable for other non-linear problems and other discretization methods, and whereas perhaps outperformed by mesh-adaptive techniques, adaptive integration requires much less implementation and computation. Both techniques can moreover be easily combined. Copyright © 2009 John Wiley & Sons, Ltd.
jueves, 8 de octubre de 2009
Adaptive Kronrod-Patterson integration of non-linear finite element matrices
Efficient simulation of unsaturated moisture flow in porous media is of great importance in many engineering fields. The highly non-linear character of unsaturated flow typically gives sharp moving moisture fronts during wetting and drying of materials with strong local moisture permeability and capacity variations as result. It is shown that these strong variations conflict with the common preference for low-order numerical integration in finite element simulations of unsaturated moisture flow: inaccurate numerical integration leads to errors that are often far more important than errors from inappropriate discretization.In response, this article develops adaptive integration, based on nested Kronrod-Patterson-Gauss integration schemes: basically, the integration order is adapted to the locally observed grade of non-linearity. Adaptive integration is developed based on a standard infiltration problem, and it is demonstrated that serious reductions in the numbers of required integration points and discretization nodes can be obtained, thus significantly increasing computational efficiency. The multi-dimensional applicability is exemplified with two-dimensional wetting and drying applications.While developed for finite element unsaturated moisture transfer simulation, adaptive integration is similarly applicable for other non-linear problems and other discretization methods, and whereas perhaps outperformed by mesh-adaptive techniques, adaptive integration requires much less implementation and computation. Both techniques can moreover be easily combined. Copyright © 2009 John Wiley & Sons, Ltd.
A POD goal-oriented error measure for mesh optimization
The approach for designing an error measure to guide an adaptive meshing algorithm proposed in Power et al. (Ocean Modell. 2006; 15:3-38) is extended to use a POD adjoint-based method, thus facilitating efficient primal and adjoint integration in time. The aim is to obtain a new mesh that can adequately resolve all the fields at all time levels, with optimal (w.r.t. the functional) efficiency. The goal-based method solves both the primal and adjoint equations to form the overall error norms, in the form of a metric tensor. The tetrahedral elements are then optimized so that they have unit size in Riemannian space defined with respect to the metric tensor.This is the first attempt to use POD to estimate an anisotropic error measure. The metric tensor field can be used to direct anisotropic mesh adaptivity. The resulting mesh is optimized to efficiently represent the solution fields over a given time period. The calculation of the error measures is carried out in the reduced space. The POD approach facilitates efficient integration backwards in time and yields the sensitivity analysis necessary for the goal-based error estimates. The accuracy of both the primal and adjoint-reduced models is thus optimized (through the use of anisotropic mesh adaptivity). In addition, the functional for optimizing meshes has been designed to be consistent with that for 4D Var data assimilation. Copyright © 2009 John Wiley & Sons, Ltd.
Relaxation Lax-Friedrichs sweeping scheme for static Hamilton-Jacobi equations
Abstract This paper presents a relaxation Lax-Friedrichs sweeping scheme to approximate viscosity solutions of static Hamilton Jacobi
equations in any number of spatial dimensions. It is a generalization of the scheme proposed in Kao et al. (J Comput Phys
196:367–391, 2004). Numerical examples suggest that the relaxation Lax-Friedrichs sweeping scheme has smaller number of iterations than the
original Lax-Friedrichs sweeping scheme when the relaxation factor ω is slightly larger than one. And first order convergence is also demonstrated by numerical results. A theoretical analysis
for our scheme in a special case is given.
equations in any number of spatial dimensions. It is a generalization of the scheme proposed in Kao et al. (J Comput Phys
196:367–391, 2004). Numerical examples suggest that the relaxation Lax-Friedrichs sweeping scheme has smaller number of iterations than the
original Lax-Friedrichs sweeping scheme when the relaxation factor ω is slightly larger than one. And first order convergence is also demonstrated by numerical results. A theoretical analysis
for our scheme in a special case is given.
- Content Type Journal Article
- Category Original Paper
- DOI 10.1007/s11075-009-9337-5
- Authors
- Peng Zhu, Hunan University College of Mathematics and Econometrics Changsha Hunan 410082 China
- Shuzi Zhou, Hunan University College of Mathematics and Econometrics Changsha Hunan 410082 China
- Journal Numerical Algorithms
- Online ISSN 1572-9265
- Print ISSN 1017-1398
Numerical studies on soil moisture distributions in heterogeneous catchments
[1] The paper deals with numerical studies of basin-scale dynamics of soil moisture in arbitrarily heterogeneous conditions (i.e., in presence of heterogeneity of climate, soil, vegetation and land use). Its relevance stems from comparative analysis of the probabilistic structure of spatially averaged soil moisture fields with the corresponding exact solutions of the underlying simplified stochastic point processes. The probabilistic structure of coarse-grained soil moisture fields is largely controlled by temporal fluctuations of intermittent rainfall fields. Averaged properties are also affected by heterogeneous soil and vegetation features and by the spatial scale of aggregation. Here, we employ results from extended Montecarlo simulations of a continuous model of the hydrologic response that proved suitable to describe observed events. The comparison of numerically derived soil moisture probability density functions with exact simplified solutions suggests, somewhat unexpectedly, that the analytical model can reasonably describe the large-scale behavior of spatially-averaged hydrologic fluxes through physically meaningful, basin-scale soil and vegetation parameters. The application of a seasonally variable, stochastic climate model shows pronounced daily fluctuations in the relationship between water losses and soil moisture, related to the underlying climatic fluctuations. The resulting spatially averaged soil moisture probability density functions in heterogeneous catchments, however, do not show appreciable differences with respect to the ones obtained assuming constant mean climate conditions. We thus conclude that effective basin-scale states, which average highly heterogeneous (spatial/temporal) properties allowing exact probabilistic descriptions, indeed exist, with implications for large scale estimates of soil-atmosphere interactions.
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