A Resourceful Splitting Technique with Applications to Deterministic and Stochastic Multiscale Finite Element Methods

For a regional land subsidence problem, there are difficulties for the traditional finite element method to solve the regional groundwater flow equation involved. A new finite element method - multiscale finite element method (MsFEM) applied to resolve three dimensional groundwater flow problem- is introduced. Three advantages make the MsFEM very useful for groundwater flow simulation in a large region with highly heterogeneous porous media. One advantage is that this method can significantly save computing efforts by coarse grid mesh. The variation of parameters in an element is brought to the multiscale base functions. Second one is that this method can to a large extent avoid large aspect ratios of elements by combining aquifers and aquitards in one element. The third advantage is that the hydraulic heads in different depths of aquitards or aquifers could be rather accurately interpolated using the multiscale base functions and the nodal heads of the element by MsFEM. The third advantage is useful for the subsidence modeling. A numerical example is done to verify the advantages of MsFEM. Finally, the MsFEM is applied to solve the regional land subsidence model of Shanghai, which includes a three-dimensional groundwater flow mathematical model and a one-dimensional subsidence model. The traditional two-step approach is followed in this study, with the hydrodynamics of the pumped multiaquifer system first simulated by a 3-D groundwater flow model and the subsidence then computed with a 1-D subsidence model with the pore pressure field specified as an external distributed source of strength within the porous medium. The simulation period is from March, 1986 to December, 1998. Using MsFEM, the aquifer system in Shanghai is divided into six layers, which means one aquifer and one aquitard are in one layer with an average thickness of about 50 meters. Every element in MsFEM is subdivided into 7 sub-layers in the vertical direction and every horizontal layer in an element is divided into 4 hexahedral elements. So each element is subdivided into 28 hexahedral sub-cells, which allows the heterogeneity of aquifers and aquitards to be considered by multiscale base functions. The aquifers system in total is discretized into 19,996 elements with a total of 25,000 nodes. The hydraulic heads in any depth of aquitards or aquifers could be interpolated using the multiscale base functions and the nodal heads of the element. It is very useful for the settlement calculation for the aquitards or aquifers after hydraulic heads at nodes of elements are obtained by MsFEM. The application of MsFEM significantly saves computing efforts, effectively decreases the aspect ratios of elements and obtains good results, which proves the MsFEM can be applied to solve the real regional 3D groundwater flow problems, and it is an effective numerical method for regional land subsidence modeling.

An iteratively adaptive multi-scale finite element method for elliptic PDEs with rough coefficients

Spatial variability of Darcy velocity is presented due to the heterogeneity of aquifer parameters. The uncertainty qualification of velocity suffers great challenge in the complex porous media. This work focuses on the use of sparse grid collocation method in velocity simulation. Since the sparse grid collocation method provides a non-intrusive way to incorporate any existing deterministic solver, the mixed finite element method is combined as the deterministic solver to retain the local continuity of Darcy velocity. We decompose the error of the velocity into three components, and illustrate that the Karhunen-Loeve truncation brings more error into velocity approximation than into head. The convergence properties of velocity moments restrict the application of sparse grid collocation method in the problems with small correlation lengths. This work provides insights towards the application of sparse grid collocation method to velocity modeling. It is demonstrated that for which problems the sparse grid collocation method is expected to be competitive with the Monte Carlo simulation. Further work about the anisotropic sparse grid collocation method should be extended to circumvent the obstacle of dimensionality.

When solving the 3D direct current resistivity logging problem with complex heterogeneous structures, the finite-element method requires the construction of a fine discretized mesh to accurately reflect the significant heterogeneity of the steel casing and the complexity of the underground structure. This fine mesh division will require a large amount of computing resources, making it difficult to solve in practice. The multiscale finite-element method can significantly reduce the size of the coefficient matrix, which greatly decreases the computational cost. However, the low-order multiscale finite-element method cannot accurately transfer the heterogeneous information of the fine mesh to the coarse mesh, which leads to numerical errors. In addition, imposing linear boundary conditions to construct multiscale basis functions may introduce resonance error. Consequently, we implement a 3D direct current resistivity logging forward algorithm based on high-order multiscale finite elements. On the one hand, we construct boundary and internal high-order multiscale basis functions by using arbitrary-order orthogonal polynomials within local cells and then linearly combine them to form complete high-order multiscale basis functions, ensuring the numerical solution’s approximation ability and convergence accuracy. On the other hand, we construct temporary high-order multiscale basis functions on extended local cells, combined with an oversampling strategy to mitigate the error problem caused by boundary resonance effects. We simulate three different electrode configurations: pole-pole, pole-dipole, and dipole-dipole. The results indicate that our new method can approximate the fine-scale reference solution on the coarse mesh with high accuracy and significantly reduced computational time when solving the linear system.

In many applications the resolution of small-scale heterogeneities remains a significant hurdle to robust and reliable predictive simulations. In particular, while material variability at the mesoscale plays a fundamental role in processes such as material failure, the resolution required to capture mechanisms at this scale is often computationally intractable. Multiscale methods aim to overcome this difficulty through judicious choice of a subscale problem and a robust manner of passing information between scales. One promising approach is the multiscale finite element method, which increases the fidelity of macroscale simulations by solving lower-scale problems that produce enriched multiscale basis functions. In this study, we present the first work toward application of the multiscale finite element method to the nonlocal peridynamic theory of solid mechanics. This is achieved within the context of a discontinuous Galerkin framework that facilitates the description of material discontinuities and does not assume the existence of spatial derivatives. Analysis of the resulting nonlocal multiscale finite element method is achieved using the ambulant Galerkin method, developed here with sufficient generality to allow for application to multiscale finite element methods for both local and nonlocal models that satisfy minimal assumptions. We conclude with preliminary results on a mixed-locality multiscale finite element method in which a nonlocal model is applied at the fine scale and a local model at the coarse scale.

Extended multiscale finite element method based on polyhedral coarse grid elements for heterogeneous materials and structures

Extended multiscale finite element method for large deflection analysis of thin-walled composite structures with complicated microstructure characteristics

A new multiscale XFEM with projection method for interaction between macrocrack and microcracks

An adaptive local–global multiscale finite volume element method for two-phase flow simulations

We are interested in solving stochastic elliptic partial differential equations (SPDE) by sampling. Approximating statistics of the stochastic solution requires computing a large number of samples, typically using an iterative method, and accelerating the resolution of these computations presents a considerable interest. In this context, we focus on the preconditioning of domain decomposition methods. We propose a stochastic preconditioner adapted to the sampled parameter field. To prevent having to determine the preconditioner for each sample, we use its truncated Karhunen-Loève (KL) expansion. The stochastic preconditioner is constructed as a surrogate of the original system based on Polynomial Chaos expansions in the dominant KL random variables. The preconditioner is computed off-line, in a preprocessing stage, and subsequently evaluated during the sampling stage (in-line). In this work, we present a review of two main contributions, which construct stochastic preconditioners in different contexts. In the first contribution (Reis et al., Preconditioning the Schwartz Method for Stochastic Elliptic Equations, Submitted), we consider an overlapping setting and construct a stochastic preconditioner for accelerating the additive Schwarz method. In the second contribution (Reis et al., Acceleration of Domain Decomposition Methods for Stochastic Elliptic Equations, Submitted), we consider a nonoverlapping setting and construct a Schur complement system. The stochastic preconditioner is used to accelerate a Conjugate Gradient method used to solve the Schur system. In both cases, numerical experiments demonstrate the significant acceleration provided by the stochastic preconditioner, compared to deterministic preconditioners such as the one based on the mean or median of the coefficient parameter.

In this article, we propose and analyse a sparse grid collocation method to solve an optimal control problem involving an elliptic partial differential equation with random coefficients and forcing terms. The input data are assumed to be dependent on a finite number of random variables. We prove that an optimal solution exists, and derive an optimality system. A Galerkin approximation in physical space and a sparse grid collocation in the probability space is used. Error estimates for a fully discrete solution using an appropriate norm are provided, and we analyse the computational efficiency. Computational evidence complements the present theory, to show the effectiveness of our stochastic collocation method.

This paper presents non-intrusive, efficient stochastic approaches for predicting uncertainties associated with petroleum reservoir simulations. The Monte Carlo simulation method, which is the most common and straightforward approach for uncertainty quantification in the industry, requires to perform a large number of reservoir simulations and is thus computationally expensive especially for large-scale problems. We propose an efficient and accurate alternative through the collocation-based stochastic approaches. The reservoirs are considered to exhibit randomly heterogeneous flow properties. The underlying random permeability field can be represented by the Karhunen-Loeve expansion (or principal component analysis), which reduces the dimensionality of random space. Two different collocation-based methods are introduced to propagate uncertainty of the reservoir response. The first one is the probabilistic collocation method that deals with the random reservoir responses by employing the orthogonal polynomial functions as the bases of the random space and utilizing the collocation technique in the random space. The second one is the sparse grid collocation method that is based on the multi-dimensional interpolation and high-dimensional quadrature techniques. They are non-intrusive in that the resulting equations have the exactly the same form as the original equations and can thus be solved with existing reservoir simulators. These methods are efficient since only a small number of simulations are required and the statistical moments and probability density functions of the quantities of interest in the oil reservoirs can be accurately estimated. The proposed approaches are demonstrated with a 3D reservoir model originating from the 9th SPE comparative project. The accuracy, efficiency, and compatibility are compared against Monte Carlo simulations. This study reveals that, compared to traditional Monte Carlo simulations, the collocation-based stochastic approaches can accurately quantify uncertainty in petroleum reservoirs and greatly reduce the computational cost.

We present an enhancement of the multiscale finite element method (MsFEM) by combining it with the hp-adaptive FEM. Such a discretization-based homogenization technique is a versatile tool for modeling heterogeneous materials with fast oscillating elasticity coefficients. No assumption on periodicity of the domain is required. In order to avoid direct, so-called overkill mesh computations, a coarse mesh with effective stiffness matrices is used and special shape functions are constructed to account for the local heterogeneities at the micro resolution. The automatic adaptivity (hp-type at the macro resolution and h-type at the micro resolution) increases efficiency of computation. In this paper details of the modified MsFEM are presented and a numerical test performed on a Fichera corner domain is presented in order to validate the proposed approach.

The multiscale finite element method (MsFEM) combined with conventional finite element method (FEM) is proposed to solve static magnetic field in the ribbon magnetic core. Firstly, a simple 2-dimensional electrostatic problem is used to introduce the MsFEM part. The results are compared to analytical method, as well as conventional FEM. Then, an example of magneto static problem is considered for a ribbon magnetic core built sheet by sheet. Conventional FEM and MsFEM are used to compute the magneto-static field by adopting scalar magnetic potential. Both magnetic potential and magnetic flux density on a certain path are compared. It is shown that the results obtained by MsFEM agrees well with the one from conventional FEM. Moreover, MsFEM combined with FEM is potentially a general strategy for multiscale modeling of ribbon magnetic cores, which can effectively reduce the computational cost.

Conventional finite-element method for solving Helmholtz equation on highly heterogeneous media usually require very finely discretized mesh to accurately represent the medium property variations, and therefore is computationally expensive. We present a continuous Galerkin generalized multiscale finite element method for solving Helmholtz equation in media with variable velocity and mass density on coarse mesh. Instead of using polynomial basis functions, our multiscale method use multiscale functions obtained by multiplying the eigenfunctions from local spectral problems with an appropriately selected multiscale partition of unity to form the approximation space on the coarse mesh. These multiscale basis can efficiently capture the information of fine scale features of the medium without solving the problem on the fine mesh. We show by a numerical example that our multiscale method can greatly reduce the dimension of the coefficient matrix of Helmholtz equation system, as well as the computational time, with very small error.