This review covers, in a comprehensive manner, the approaches available in the literature to upscale soil water processes and hydraulic parameters in the vadose zone. We distinguish two categories of upscaling methods: forward approaches requiring information about the spatial distribution of hydraulic parameters at a small scale, and inverse modeling approaches requiring information about the spatial and temporal variation of state variables at various scales, including so-called “soft data”. Geostatistical and scaling approaches are crucial to upscale soil water processes and to derive large-scale effective fluxes and parameters from small-scale information. Upscaling approaches include stochastic perturbation methods, the scaleway approach, the stream-tube approach, the aggregation concept, inverse modeling approaches, and data fusion. With all upscaling methods, the estimated effective parameters depend not only on the properties of the heterogeneous flow field but also on boundary conditions. The use of the Richards equation at the field and watershed scale is based more on pragmatism than on a sound physical basis. There are practically no data sets presently available that provide sufficient information to extensively validate existing upscaling approaches. Use of numerical case studies has therefore been most common. More recently and still under development, hydrogeophysical methods combined with ground-based remote sensing techniques promise significant contributions toward providing high-quality data sets. Finally, most of the upscaling literature in vadose zone research has dealt with bare soils or deep vadose zones. There is a need to develop upscaling methods for real world soils, considering root water uptake mechanisms and other soil–plant–atmosphere interactions.

Upscaling Hydraulic Properties and Soil Water Flow Processes in Heterogeneous Soils: A Review

A strong coupling partitioned approach for fluid–structure interaction with free surfaces

The topological asymptotic analysis provides the sensitivity of a given shape functional with respect to an infinitesimal domain perturbation. Therefore, this sensitivity can be naturally used as a descent direction in a structural topology design problem. However, according to the literature concerning the topological derivative, only the classical approach based on flexibility minimization for a given amount of material, without control on the stress level supported by the structural device, has been considered. In this paper, therefore, we introduce a class of penalty functionals that mimic a pointwise constraint on the Von Mises stress field. The associated topological derivative is obtained for plane stress linear elasticity. Only the formal asymptotic expansion procedure is presented, but full justifications can be deduced from existing works. Then, a topology optimization algorithm based on these concepts is proposed, that allows for treating local stress criteria. Finally, this feature is shown through some numerical examples.

Topological optimization of structures subject to Von Mises stress constraints

We discuss the numerical solution of structured generalized eigenvalue problems that arise from linear-quadratic optimal control problems, $H_{\infty}$ optimization, multibody systems, and many other areas of applied mathematics, physics, and chemistry. The classical approach for these problems requires computing invariant and deflating subspaces of matrices and matrix pencils with Hamiltonian and/or skew-Hamiltonian structure. We extend the recently developed methods for Hamiltonian matrices to the general case of skew-Hamiltonian/Hamiltonian pencils. The algorithms circumvent problems with skew-Hamiltonian/Hamiltonian matrix pencils that lack structured Schur forms by embedding them into matrix pencils that always admit a structured Schur form. The rounding error analysis of the resulting algorithms is favorable. For the embedded matrix pencils, the algorithms use structure-preserving unitary matrix computations and are strongly backwards stable, i.e., they compute the exact structured Schur form of a nearby matrix pencil with the same structure.

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Numerical Computation of Deflating Subspaces of Skew-Hamiltonian/Hamiltonian Pencils

We consider boundary value problems for elliptic systems in a domain complementary to a smooth surface with boundary, which models a crack with its edge. The same boundary conditions are prescribed...

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Asymptotics Without Logarithmic Terms for Crack Problems

This paper is devoted to some transmission problems for the Laplace and linear elasticity operators in two- and three-dimensional nonsmooth domains. We investigate the behaviour of harmonic and linear elastic fields near geometrical singularities, especially near corner points or edges where the interface intersects with the boundaries. We give a short overview about the known results for 2-D problems and add new results for 3-D problems. Numerical results for the calculation of the singular exponents in the asymptotic expansion are presented for both two- and three-dimensional problems. Some spectral properties of the corresponding parameter depending operator bundles are also given. Furthermore, we derive boundary integral equations for the solution of the transmission problems, which lead finally to "local" pseudo-differential operator equations with corresponding Steklov–Poincare operators on the interface. We discuss their solvability and uniqueness. The above regularity results are used in order to characterize the regularity of the solutions of these integral equations.

Transmission problems for the laplace and elasticity operators: regularity and boundary integral formulation

The Many Roles of Viscosity in Solid Mechanics.- Multiscale Approach to a Basic Problem of Materials Mechanics (Propagation of Phase-Transition Fronts).- Section 1: Two and Multiphase Flows.- Analysing Particle Sedimentation in Fluids by Measure-Valued Stochastic Processes.- Phenomenological Model of Sedimentation-Consolidation Processes.- The Structure of Bidisperse Suspensions at Low Reynolds Numbers.- Lattice Boltzmann Simulations of Complex Multiphase Flows.- Interface Tracking in Multiphase Flows.- Section 2: Mechanics of Materials and Multiscaling.- Finite Element Computation of Macroscopic Quantities in Nonconvex Minimisation Problems and Applications in Materials Science.- Homogenization of an Initial-Boundary Value Problem Describing Evolving Microstructure.- Aspects of Homogenization Techniques and Multigrid Solving.- Estimates on the Mixture Function for Multiphase Problems in Elasticity.- A Two-Scale Micro-Macro-Approach to Anisotropic Finite Plasticity of Polycrystals.- Section 3: Solid-Fluid-Interaction.- Boundary Conditions at the Viscous Sliding Interface of Incompressible Porous Deformable Media.- Computational Experience from the Solution of Coupled Problems in Ship Dynamics.- On the Adaptive Computation of Shear Bands in Frictional Geomaterials.- Intrinsic Viscoelasticity of Porous Materials.- Numerical Simulation of Fluids Interacting with Moving Rigid Bodies.- Partitioned Analysis of Transient Nonlinear Fluid Structure Interaction Problems Including Free Surface Effects.- Section 4: Efficient Solvers and Adaptivity.- Domain Decomposition Methods in the Design of High Power Electronic Devices.- Matrix Compression for the Radiation Heat Transfer in Exhaust Pipes.- An Iterative Substructuring Method with Lagrange Multipliers for Elasticity Problems Using Approximate Neumann Subdomain Solvers.- A New a Posteriori Error Estimator in Adaptive Direct Boundary Element Methods. The Neumann Problem.- Efficient Elasto-Plastic Simulation.- Section 5: Contact and Fracture.- Sensitivity and Optimal Control in Contact Mechanics.- On the Treatment of Contact Problems in Elasto-Plasticity.- On Contact Problems for Linear and Nonlinear Elastic Plates: Existence of Solutions and Application of Augmented Lagrangian Method.- Longitudinal Wave Propagation in Conical Rods Subject to Impacts.- A Survey on Dynamic Contact Problems with Coulomb Friction.- Numerical Simulation of Noise Radiation from Rolling Tires.- Stress Singularities in Composites.

Multifield problems : state of the art

The linear model equations of elasticity often give rise to oscillatory solutions in some vicinity of interface crack fronts. In this paper we apply the Wiener-Hopf method which yields the asymptotic behaviour of the elastic fields and, in addition, criteria to prevent oscillatory solutions. The exponents of the asymptotic expansions are found as eigenvalues of the symbol of corresponding boundary pseudodifferential equations. The method works for three-dimensional anisotropic bodies and we demonstrate it for the example of two anisotropic bodies, one of which is bounded and the other one is its exterior complement. The common boundary is a smooth surface. On one part of this surface, called the interface, the bodies are bonded, while on the complementary part there is a crack. By applying the potential method, the problem is reduced to an equivalent system of Boundary Pseudodifferential Equations (BPE) on the interface with the stress vector as the unknown. The BPEs are defined via Poincare-Steklov operators. We prove the unique solvability of these BPEs and obtain the full asymptotic expansion of the solution near the crack front. As a special case we consider the interface crack between two different isotropic materials and derive an explicit criterion which prevents oscillatory solutions.

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Interface cracks in anisotropic composites

It is well known that high stress concentrations can occur in elastic composites in particular due to the interaction of geometrical singularities like corners, edges and cracks and structural singularities like jumping material parameters. In the project C5 Stress concentrations in heterogeneous materials of the SFB 404 it was mathematically analyzed where and which kind of stress singularities in coupled linear and nonlinear elastic structures occur. In the linear case asymptotic expansions near the geometrical and structural peculiarities are derived, formulae for generalized stress intensity factors included. In the nonlinear case such expansions are unknown in general and regularity results are proved for elastic materials with power-law constitutive equations with the help of the difference quotient technique combined with a quasi-monotone covering condition for the subdomains and the energy densities. Furthermore, some applications of the regularity results to shape and structure optimization and the Griffith fracture criterion in linear and nonlinear elastic structures are discussed. Numerical examples illustrate the results.

Regularity of Elastic Fields in Composites

Material discontinuities and geometrical peculiarities like cracks, notches, corners and edges lead to stress singularities in linear elastic structures. Their strength can be characterised by stress intensity or notch factors. In this paper we investigate the influence of the shape and of the material parameters on these factors. The sensitivity analysis is performed by means of asymptotic expansions.

A.-M. Sändig

Papers

Transmission problems for the laplace and elasticity operators: regularity and boundary integral formulation

Multifield problems : state of the art

Interface cracks in anisotropic composites

Regularity of Elastic Fields in Composites

Sensitivity analysis of stress intensity factors in 2D coupled elastic structures