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Schur preconditioning of the Stokes equations in channel-dominated domains
Abstract
A novel precondition operator for the Schur complement of the stationary Stokes equations is proposed. Numerical experiments demonstrate that its discrete version is superior to established precondition operators if the computational domain is a thin channel or contains such (e.g. porous media or filters). Discrete diffusion is added to the established precondition operator, which is heuristically motivated by Darcy’s law—a homogenization result of Stokes’ equations on perforated domains. As proof of concept, we embed the operator into a block-diagonal precondition matrix for MINRES and solve the stationary Stokes saddle-point system. For domains with complex geometries, the number of iterations required to meet a certain tolerance is significantly reduced. The proposed operator can easily be incorporated into an already existing numerical solver, does not change the sparsity pattern of the established precondition operator for finite element discretizations, and is in particular suitable for distributed parallel implementations.
... Keyes et al. (2013) demonstrates that the Schur complement method can precondition systems for solving multiphysics problems in common hybrid approaches such as the Newton-Krylov method. Numerous studies have explored the use of Schur complement method for preconditioning approaches (Dassi and Scacchi, 2020;Franceschini et al., 2019;Meier et al., 2022;Dorostkar et al., 2016;Benner et al., 2016;Little et al., 2003;Murphy et al., 2000;He et al., 2018;Kraus, 2012;Dillon et al., 2018;Dener and Hicken, 2017). The Schur complement method has also been used to solve problems arising from hybridized and embedded discontinuous Galerkin methods (Fidkowski and Chen, 2019;Bai and Fidkowski, 2022). ...
Simulation-based multiphysics and multidisciplinary models are fundamental building blocks of design optimization frameworks that involve coupled systems. One common challenge arises when the coupled linear and nonlinear systems that represent these models result in a saddle-point problem. Such problems require a Newton-type method instead of block Gauss-Seidel (BGS) based methods because the Jacobian matrix has a non-invertible block. We introduce nonlinear Schur complement (NSC) and linear Schur complement (LSC) solvers suitable for solving saddle-point problems. We implement these solvers in NASA's OpenMDAO framework and demonstrate their effectiveness through two analytic problems and an aerodynamic shape optimization of an aircraft wing. The solvers enable flexible and robust formulation of optimization problems by circumventing the numerical challenges inherent in saddle-point systems. This approach can be applied to a wide range of saddle-point problems in simulation-based optimization, making it particularly valuable for multidisciplinary design applications.
... One possible solution for solving large-scale saddle-point problems is the Schur-complement (SC) method, which is a decoupled approach. There have been significant efforts to use SC method for preconditioning approaches [4][5][6][7][8][9][10][11][12][13][14]. However, only a few studies have addressed the saddle-point problems in multidisciplinary models [2,15]. ...
Multidisciplinary models are composed of numerous implicit systems that are solved in a coupled manner. A common challenge arises when an individual discipline’s Jacobian is non-invertible, which results in the coupled multidisciplinary model representing a saddle-point problem. In general, the small-scale multidisciplinary saddle-point problems are solved with the coupled Newton’s method. Nevertheless, coupled solver approaches are difficult to build and have robustness problems. In addition, they necessitate solving large coupled linear systems in large-scale applications. In this work, we use the newly developed nonlinear and linear Schur complement (SC) solvers, appropriate for multidisciplinary models based on computational fluid dynamics (CFD) to solve these saddle-point problems. The SC solvers are not susceptible to the robustness problems of conventional coupled solutions since they leverage specialized CFD solvers. Thanks to these solvers, the conventional constrained optimization, which has a saddle-point system in its Jacobian, can be transformed into unconstrained optimizations. In this work, we demonstrate several applications of Schur-complement-based optimizations of CFD-based saddle-point problems: aerodynamic shape optimization (ASO) of a wing and coupled aeropropulsive design optimization of a podded propulsor. In the ASO cases, the Schur complement solvers-based optimizations outperform the conventional constrained optimizations. Although the conventional optimization in the coupled aeropropulsive design optimization problem outperforms the SC solvers-based optimization, it requires an effective scaling of the design variables of the boundary conditions and the constraints. SC solvers-based optimizations always provide a feasible design at each design iteration, whereas the conventional approach does not. Furthermore, in large-scale nonlinear saddle-point problems, the nonlinear SC solver offers an alternative way to obtain a feasible solution for a given design instead of carrying out a modest optimization in order to get feasible solutions. In the CFD-based saddle-point applications, SC solvers will play an important role in the future because they are robust and can solve the saddle-point problem using the native solvers of the specialized CFD model.
... Recently, it was demonstrated in [15] that adding discrete diffusion to the established preconditioner significantly reduces the number of iterations in the case of channel-dominated domains. Similar results were demonstrated in [20] for the diffusion-like SIMPLE preconditioner in the case of complex geometries from tight porous media. ...
If the Stokes equations are properly discretized, it is well-known that the Schur complement matrix is spectrally equivalent to the identity matrix. Moreover, in the case of simple geometries, it is often observed that most of its eigenvalues are equal to one. These facts form the basis for the famous Uzawa and Krylov-Uzawa algorithms. However, in the case of complex geometries, the Schur complement matrix can become arbitrarily ill-conditioned having a significant portion of non-unit eigenvalues, which makes the established Uzawa preconditioner inefficient. In this article, we study the Schur complement formulation for the staggered finite-difference discretization of the Stokes problem in 3D CT images and synthetic 2D geometries. We numerically investigate the performance of the CG iterative method with the Uzawa and SIMPLE preconditioners and draw several conclusions. First, we show that in the case of low porosity, CG with the SIMPLE preconditioner converges faster to the discrete pressure and provides a more accurate calculation of sample permeability. Second, we show that an increase in the surface-to-volume ratio leads to an increase in the condition number of the Schur complement matrix, while the dependence is inverse for the Schur complement matrix preconditioned with the SIMPLE. As an explanation, we conjecture that the no-slip boundary conditions are the reason for non-unit eigenvalues of the Schur complement.
We discuss the development of a two-level iterative solver for Stokes flow simulations in porous media, focusing on its application to compute absolute permeability from pixel-based microstructures. The Schur complement plays a key role in converting the saddle-point system arising from the weak form of the Stokes equations into a condensed system for velocity, solvable using a robust and efficient two-level solver. At the first (external) level, we employ the conjugate gradient method, though this choice is not restrictive, as alternatives like the minimal residual method can also be accommodated. As in many short-recurrence Krylov subspace methods, the global matrix of the condensed system does not need to be stored in memory, as long as a function to compute matrix–vector products is available. The inverse matrix from the Schur complement is replaced by subsystem solutions during the implementation of matrix–vector products, similarly to some preconditioning techniques. These subsystems define the second (internal) level, which can also be handled iteratively. In addition, the proposed solver is designed within the framework of image-based simulations via the finite element method, making it suitable for memory-efficient, matrix-free implementations on structured meshes. We demonstrate the solver’s applicability through numerical results on a range of image-based porous media characterization problems, observing stable and fast convergence patterns in terms of iterations. Among these, we specifically apply the solver to characterize two distinct porous media imaged via X-ray micro-computed tomography: a fiber-reinforced composite and a sandstone sample.
It is known (see, e.g., [SIAM J. Matrix Anal. Appl. 2014;35(1):143‐173]) that the performance of iterative methods for solving the Stokes problem essentially depends on the quality of the preconditioner for the Schur complement matrix, . In this paper, we consider two preconditioners for : the identity one and the SIMPLE one, and numerically study their performance for solving the Stokes problem in tight geometries. The latter are characterized by a high surface‐to‐volume ratio. We show that for such geometries, can become severely ill‐conditioned, having a very large condition number and a significant portion of non‐unit eigenvalues. As a consequence, the identity matrix, which is broadly used as a preconditioner for solving the Stokes problem in simple geometries, becomes very inefficient. We show that there is a correlation between the surface‐to‐volume ratio and the condition number of : the latter increases with the increase of the former. We show that the condition number of the diffusive SIMPLE‐preconditioned Schur complement matrix remains bounded when the surface‐to‐volume ratio increases, which explains the robust performance of this preconditioner for tight geometries. Further on, we use a direct method to calculate the full spectrum of and show that there is a correlation between the number of its non‐unit eigenvalues and the number of grid points at which no‐slip boundary conditions are prescribed. To illustrate the above findings, we examine the Pressure Schur Complement formulation for staggered finite‐difference discretization of the Stokes equations and solve it with the preconditioned conjugate gradient method. The practical problem which is of interest to us is computing the permeability of tight rocks.
The intrinsic permeability is a crucial parameter to characterise and quantify fluid flow through porous media. However, this parameter is typically uncertain, even if the geometry of the pore structure is available. In this paper, we perform a comparative study of experimental, semi-analytical and numerical methods to calculate the permeability of a regular porous structure. In particular, we use the Kozeny–Carman relation, different homogenisation approaches (3D, 2D, very thin porous media and pseudo 2D/3D), pore-scale simulations (lattice Boltzmann method, Smoothed Particle Hydrodynamics and finite-element method) and pore-scale experiments (microfluidics). A conceptual design of a periodic porous structure with regularly positioned solid cylinders is set up as a benchmark problem and treated with all considered methods. The results are discussed with regard to the individual strengths and limitations of the used methods. The applicable homogenisation approaches as well as all considered pore-scale models prove their ability to predict the permeability of the benchmark problem. The underestimation obtained by the microfluidic experiments is analysed in detail using the lattice Boltzmann method, which makes it possible to quantify the influence of experimental setup restrictions.
Conductive structured catalysts offer significant potential for the intensification of gas-solid catalytic processes owing to their enhanced heat transfer properties. A major drawback is the limited catalyst inventory. Recently, packed foams were proposed to overcome the catalyst inventory limitation. The effectiveness of this concept was proven at lab-scale for intensified reactors filled with small catalyst particles. When adopting commercial foams and industrial-scale catalyst pellets, however, poor packing efficiencies are expected, limiting the potential of this concept.
Similarly to foams, Periodic Open Cellular Structures (POCS) grant high heat transfer rates thanks to substantial heat conduction in their solid matrix. Additively manufactured POCS additionally offer great design flexibility. This allows for using a wider range of pellet sizes. In this work, particle packed POCS are introduced and packing efficiencies are systematically studied. Pressure drop in packed POCS is also analyzed and a suitable correlation is proposed. The heat transfer associated with this innovative reactor solution is investigated by performing non-reactive heat transfer experiments. Based on these experiments, a predictive heat transfer model is established and successfully validated with experimental data. The enormous potential of packed POCS for process intensification is illustrated by a case study of a Sabatier pilot reactor.
This paper is concerned with the implementation of efficient solution algorithms for elliptic problems with constraints. We establish theory which shows that including a simple scaling within well-established block diagonal preconditioners for Stokes problems can result in significantly faster convergence when applying the preconditioned MINRES method. The codes used in the numerical studies are available online.
We homogenize stationary incompressible Stokes flow in a periodic porous medium. The fluid is assumed to satisfy a no-slip condition on the boundary of solid inclusions and a normal stress (traction) condition on the global boundary. Under these assumptions, the homogenized equation becomes the classical Darcy law with a Dirichlet condition for the pressure.
We present a novel numerical scheme for solving the Stokes equation with variable coecients in the unit box. Our scheme is based on a volume integral equation formulation. Compared to finite element methods, our formulation de-couples the velocity and pressure, generates velocity fields that are by construction divergence free to high accuracy and its performance does not depend on the order of the basis used for discretization. In addition, we employ a novel adaptive fast multipole method for volume integrals to obtain a scheme that is algorithmically optimal. Our scheme supports non-uniform discretizations and is spectrally accurate. To increase per node performance, we have integrated our code with both NVIDIA and Intel accelerators. In our largest scalability test, we solved a problem with 20 billion unknowns, using a 14-order approximation for the velocity, on 2048 nodes of the Stampede system at the Texas Advanced Computing Center. We achieved 0.656 petaFLOPS for the overall code (23% eciency) and one petaFLOPS for the volume integrals (33% eciency). As an application example, we simulate Stokes flow in a porous medium with highly complex pore structure using a penalty formulation to enforce the no slip condition.
In this work, a complete work flow from pore-scale imaging to absolute permeability determination is described and discussed. Two specific points are tackled, concerning (1) the mesh refinement for a fixed image resolution and (2) the impact of the determination method used. A key point for this kind of approach is to work on enough large samples to check the representativity of the obtained evaluations, which requires efficient parallel capabilities. Image acquisition and processing are realized using a commercial micro-tomograph. The pore-scale flows are then evaluated using the finite volume method implemented in the open-source platform OpenFOAM ® . For this numerical method, the influence of the different aspects mentioned above are studied. Moreover, the parallel efficiency is also tested and discussed. We observe that the level of mesh refinement has a non-negligible impact on permeability tensor. Moreover, increasing the refinement level tends to reduce the gap between the methods of computational measurements. The increase in computation time with the mesh is balanced with the good parallel efficiency of the platform.
Pore-scale imaging and modelling - digital core analysis - is becoming a
routine service in the oil and gas industry, and has potential
applications in contaminant transport and carbon dioxide storage. This
paper briefly describes the underlying technology, namely imaging of the
pore space of rocks from the nanometre scale upwards, coupled with a
suite of different numerical techniques for simulating single and
multiphase flow and transport through these images. Three example
applications are then described, illustrating the range of scientific
problems that can be tackled: dispersion in different rock samples that
predicts the anomalous transport behaviour characteristic of highly
heterogeneous carbonates; imaging of super-critical carbon dioxide in
sandstone to demonstrate the possibility of capillary trapping in
geological carbon storage; and the computation of relative permeability
for mixed-wet carbonates and implications for oilfield waterflood
recovery. The paper concludes by discussing limitations and challenges,
including finding representative samples, imaging and simulating flow
and transport in pore spaces over many orders of magnitude in size, the
determination of wettability, and upscaling to the field scale. We
conclude that pore-scale modelling is likely to become more widely
applied in the oil industry including assessment of unconventional oil
and gas resources. It has the potential to transform our understanding
of multiphase flow processes, facilitating more efficient oil and gas
recovery, effective contaminant removal and safe carbon dioxide storage.
A new generation, parallel adaptive-mesh mantle convection code, Rhea, is described and benchmarked. Rhea targets large-scale mantle convection simulations on parallel computers, and thus has been developed with a strong focus on computational efficiency and parallel scala-bility of both mesh handling and numerical solvers. Rhea builds mantle convection solvers on a collection of parallel octree-based adaptive finite element libraries that support new distributed data structures and parallel algorithms for dynamic coarsening, refinement, rebalancing and repartitioning of the mesh. In this study we demonstrate scalability to 122 880 compute cores and verify correctness of the implementation. We present the numerical approximation and convergence properties using 3-D benchmark problems and other tests for variable-viscosity Stokes flow and thermal convection.
The LBB condition is well-known to guarantee the stability of a finite
element (FE) velocity - pressure pair in incompressible flow calculations.
To ensure the condition to be satisfied a certain constant should be positive and
mesh-independent. The paper studies the dependence of the LBB condition on the
domain geometry. For model domains such as strips and rings the
substantial dependence of this constant on geometry aspect ratios is observed.
In domains with highly anisotropic substructures this may require special care
with numerics to avoid failures similar to those when
the LBB condition is violated. In the core of the paper
we prove that for any FE velocity-pressure pair satisfying usual approximation
hypotheses the mesh-independent limit in the LBB condition is not greater than
its continuous counterpart, the constant from the Nečas inequality.
For the latter the explicit and asymptotically accurate estimates are proved.
The analytic results are illustrated by several numerical experiments.
The hypre software library provides high performance preconditioners and solvers for the solution of large, sparse linear systems on
massively parallel computers. One of its attractive features is the provision of conceptual interfaces. These interfaces give application users a more natural means for describing their linear systems, and provide access to
methods such as geometric multigrid which require additional information beyond just the matrix. This chapter discusses the
design of the conceptual interfaces in hypre and illustrates their use with various examples. We discuss the data structures and parallel implementation of these interfaces.
A brief overview of the solvers and preconditioners available through the interfaces is also given.
The Ladyzhenskaya-Babushka-Brezzi inequality (inf-sup-condition) is often used in the analysis of convergence of approximate
solutions of hydrodynamic equations to exact ones. The constant involved in it depends on the shape of the domain and determines
the efficiency of different algorithms. In this paper, its asymptotics and two-sided estimates for rectangular domains are
obtained. To this end, a new method for estimating eigenvalues of a certain spectral problem with Stokes’ saddle operator
is used.
The paper concerns with an iterative technique for solving discretized Stokes type equations with varying viscosity coe-cient. We build a special block preconditioner for the discrete system of equations and perform an analysis revealing its properties. The subject of this paper is motivated by numerical solution of incompressible non-Newtonian ∞uid equations. In particular, the general analysis is applied to the linearized equations of the regularized Bingham model of viscoplastic ∞uid. Both theoretical analysis and numerical experiments show that the suggested preconditioner leads to a signiflcant improvement of an iterative method convergence compared to a 'standard' preconditioner.
Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.
Permeability estimation of porous media from directly solving the Navier–Stokes equations has a wide spectrum of applications in petroleum industry. In this paper, we utilize a pressure-correction projection algorithm in conjunction with the interior penalty discontinuous Galerkin scheme for space discretization to build an incompressible Navier–Stokes simulator and to use this simulator to calculate permeability of real rock samples. The proposed method is accurate, numerically robust, and exhibits the potential for tackling realistic problems.
Structural hierarchy is a fundamental characteristic of natural porous media. Yet it provokes one of the grand challenges for the modeling of fluid flow and transport since pore‐scale structures and continuum‐scale domains often coincide independent of the observation scale. Common approaches to represent structural hierarchy build, for example, on a multidomain continuum for transport or on the coupling of the Stokes equations with Darcy's law for fluid flow. These approaches, however, are computationally expensive or introduce empirical parameters that are difficult to derive with independent observations. We present an efficient model for fluid flow based on Darcy's law and the law of Hagen‐Poiseuille that is parameterized based on the explicit pore space morphology obtained, for example, by X‐ray μ‐CT and inherently permits the coupling of pore‐scale and continuum‐scale domain. We used the resulting flow field to predict the transport of solutes via particle tracking across the different domains. Compared to experimental breakthrough data from laboratory‐scale columns with hierarchically structured porosity built from solid glass beads and microporous glass pellets, an excellent agreement was achieved without any calibration. Furthermore, we present different test scenarios to compare the flow fields resulting from the Stokes‐Brinkman equations and our approach to comprehensively illustrate its advantages and limitations. In this way, we could show a striking efficiency and accuracy of our approach that qualifies as general alternative for the modeling of fluid flow and transport in hierarchical porous media, for example, fractured rock or karstic aquifers.
We present a weighted BFBT approximation (w-BFBT) to the inverse Schur complement of a Stokes system with highly heterogeneous viscosity. When used as part of a Schur complement-based Stokes preconditioner, we observe robust fast convergence for Stokes problems with smooth but highly varying (up to 10 orders of magnitude) viscosities, optimal algorithmic scalability with respect to mesh refinement, and only a mild dependence on the polynomial order of high-order finite element discretizations (, order ). For certain difficult problems, we demonstrate numerically that w-BFBT significantly improves Stokes solver convergence over the widely used inverse viscosity-weighted pressure mass matrix approximation of the Schur complement. In addition, we derive theoretical eigenvalue bounds to prove spectral equivalence of w-BFBT. Using detailed numerical experiments, we discuss modifications to w-BFBT at Dirichlet boundaries that decrease the number of iterations. The overall algorithmic performance of the Stokes solver is governed by the efficacy of w-BFBT as a Schur complement approximation and, in addition, by our parallel hybrid spectral-geometric-algebraic multigrid (HMG) method, which we use to approximate the inverses of the viscous block and variable-coefficient pressure Poisson operators within w-BFBT. Building on the scalability of HMG, our Stokes solver achieves a parallel efficiency of 90% while weak scaling over a more than 600-fold increase from 48 to all 30,000 cores of TACC's Lonestar 5 supercomputer.
This paper we prove the convergence of the homogenization process of the Stokes equations with Dirichlet boundary condition in a periodic porous medium. We consider here the case where the solid part of the porous medium is connected, and we generalize to this case the results obtained by Tartar (1980).
This article presents a systematic quantitative performance analysis for large finite element computations on extreme scale computing systems. Three parallel iterative solvers for the Stokes system, discretized by low order tetrahedral elements, are compared with respect to their numerical efficiency and their scalability running on up to 786432 parallel threads. A genuine multigrid method for the saddle point system using an Uzawa-type smoother provides the best overall performance with respect to memory consumption and time-to-solution. The largest system solved on a Blue Gene/Q system has more than ten trillion (1.1⋅10^13) unknowns and requires about 13 minutes compute time. Despite the matrix free and highly optimized implementation, the memory requirement for the solution vector and the auxiliary vectors is about 200 TByte. Brandt's notion of "textbook multigrid efficiency" is employed to study the algorithmic performance of iterative solvers. A recent extension of this paradigm to "parallel textbook multigrid efficiency" makes it possible to assess also the efficiency of parallel iterative solvers for a given hardware architecture in absolute terms. The efficiency of the method is demonstrated for simulating incompressible fluid flow in a pipe filled with spherical obstacles.
Verification of Calculations involves error estimation, whereas Verification of Codes involves error evaluation, from known benchmark solutions. The best benchmarks are exact analytical solutions with sufficiently complex solution structure; they need not be realistic since Verification is a purely mathematical exercise. The Method of Manufactured Solutions (MMS) provides a straightforward and quite general procedure for generating such solutions. For complex codes, the method utilizes Symbolic Manipulation, but here it is illustrated with simple examples. When used with systematic grid refinement studies, which are remarkably sensitive, MMS produces strong Code Verifications with a theorem-like quality and a clearly defined completion point.
The computational solution of problems can be restricted by the availability of solution methods for linear(ized) systems of equations. In conjunction with iterative methods, preconditioning is often the vital component in enabling the solution of such systems when the dimension is large. We attempt a broad review of preconditioning methods.
The multi-phase flow behavior of complex rocks with broad pore size distributions often digresses from classical relations. Pore-scale simulation methods can be a great tool to improve the understanding of this behavior. However, the broad range of pore sizes present makes it difficult to gather the experimental input data needed for these simulations and poses great computational challenges. We developed a novel micro-computed-tomography (micro-CT) based dual pore network model (DPNM), which takes microporosity into account in an upscaled fashion using symbolic network elements called micro-links, while treating the macroporosity as a traditional pore network model. The connectivity and conductivity of the microporosity is derived from local information measured on micro-CT scans. Microporous connectivity is allowed both in parallel and in series to the macropore network. We allow macropores to be drained as a consequence of their connection with microporosity, permitting simulations where the macropore network alone does not percolate. The validity of the method is shown by treating an artificial network and a network extracted from a micro-CT scan of Estaillades limestone.
We consider the multigrid solution of the generalized Stokes equations with a segregated (i.e., equationwise) Gauss-Seidel smoother based on a Uzawa-type iteration. We analyze the smoother in the framework of local Fourier analysis, and obtain an analytic bound on the smoothing factor showing uniform performance for a family of Stokes problems. These results are confirmed by the numerical computation of the two-grid convergence factor for different types of grids and discretizations. Numerical results also show that the actual convergence of the W-cycle is approximately the same as that obtained by a Vanka smoother, despite this latter smoother being significantly more costly per iteration step.
In this article a numerical method for the efficient simulation of instationary, viscous, incompressible flows in 2d and 3d is described. The time-discretization is based on the fractional step `-scheme in a variant as an operator splitting, which was introduced in [3].
3D fluid dynamic numerical simulations of different structured open-cell foams are presented: a cubic lattice, the tetrakaidecahedron geometry (Kelvin cell geometry), and the diamond structure. The simulations are done with the open-source software OpenFOAM. Beneath the pressure drops of the different structures, which are compared using dimensionless numbers such as Reynolds number and Euler number, also the development of velocity and its frequency distribution, as an important parameter for chemical reactions, are evaluated. As a main result, the diamond structure provides good conditions with respect to dwell time distributions for the underlying chemical reaction. All the results give a deep knowledge in terms of the underlying principles and optimization approaches.
This article considers the iterative solution of a finite element
discretisation of magma dynamics equations. In simplified form, the magma
dynamics equations share some features of the Stokes equations. We therefore
formulate, analyse and numerically test a Elman, Silvester and Wathen-type
block preconditioner for magma dynamics. We prove analytically, and demonstrate
numerically, optimality of the preconditioner. The presented analysis
highlights the dependence of the preconditioner on parameters in the magma
dynamics equations that can affect convergence of iterative linear solvers. The
analysis is verified through a range of two- and three-dimensional numerical
examples on unstructured grids, from simple illustrate problems through to
large problems on subduction zone-like geometries. The computer code to
reproduce all numerical examples is freely available as supporting material.
3D numerical simulations of structured open-cell foams, more precisely the tetrakaidecahedron geometry (Kelvin cell geometry) are presented. The velocity and pressure field for a single cell with varied surface roughness and the whole reactor with the same pore size are calculated in order to show the influence of the surface roughness regarding the pressure drop and the influence of the length of the whole reactor regarding the velocity development, respectively. The numerical results are compared using dimensionless numbers such as Reynolds number, Euler number and friction factor.
The fundamental solution or Green’s function for flow in porous media is determined using Stokesian dynamics, a molecular-dynamics-like simulation method capable of describing the motions and forces of hydrodynamically interacting particles in Stokes flow. By evaluating the velocity disturbance caused by a source particle on field particles located throughout a monodisperse porous medium at a given value of volume fraction of solids ϕ, and by considering many such realizations of the (random) porous medium, the fundamental solution is determined. Comparison of this fundamental solution with the Green’s function of the Brinkman equation shows that the Brinkman equation accurately describes the flow in porous media for volume fractions below 0.05. For larger volume fractions significant differences between the two exist, indicating that the Brinkman equation has lost detailed predictive value, although it still describes qualitatively the behavior in moderately concentrated porous media. At low ϕ where the Brinkman equation is known to be valid, the agreement between the simulation results and the Brinkman equation demonstrates that the Stokesian dynamics method correctly captures the screening characteristic of porous media. The simulation results for ϕ≥0.05 may be useful as a basis of comparison for future theoretical work.
Preconditioners are often conceived as approximate inverses. For nonsingular indefinitematrices of saddle-point (or KKT) form, we show how preconditioners incorporating an exact Schurcomplement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues.Thus approximations of the Schur complement lead to preconditioners which can be very effective eventhough they are in no sense approximate inverses.
The method of conjugate gradients for solving systems of linear equations with a symmetric positive definite matrix A is given as a logical development of the Lanczos algorithm for tridiagonalizing A. This approach suggests numerical algorithms for solving such systems when A is symmetric but indefinite. These methods have advantages when A is large and sparse.
Following an idea of G. Nguetseng, the author defines a notion of “two-scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its “two-scale” limit, up to a strongly convergent remainder in ) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations.
For channel domains and plates the LBB constant and the corresponding eigenvalue problem of the Schur complement are investigated. Estimates for the first eigenvalue and the number of small eigenvalues are given. The numerical results confirm the theory.
We consider an algorithm for the solution of a mixed finite-element approximation of the Stokes equations in a bounded, simply
connected domain Ω ⊂ R2. The original indefinite problem for the velocity and pressure can be transformed into an equation Lp = g for the pressure
involving a symmetric, positive definite, continuous linear operator L: L2(Ω)/R → L2(Ω)/R. We apply a conjugate gradient algorithm to this equation. Each evaluation of Lp requires the solution of two discrete
Poisson equations. This is done approximately using a multigrid algorithm. The resulting iterative process has a convergence
rate κ bounded away from one independently of the meshsize. Numerical experiments yield values for κ between 0-8 and 0-93.
The generalization of the analysis to other mixed problems is obvious.
A calculation is given of the viscous force, exerted by a flowing fluid on a dense swarm of particles. The model underlying
these calculations is that of a spherical particle embedded in a porous mass. The flow through this porous mass is decribed
by a modification of Darcy's equation. Such a modification was necessary in order to obtain consistent boundary conditions.
A relation between permeability and particle size and density is obtained. Our results are compared with an experimental relation
due to Carman.
Computable bounds are established on the extremal eigenvalues of the stiffness and mass matrices. These are then applied to study the dependence of the spectral condition number of the “stiffness” and “mass” (Gram) matrices on the discretization and intrinsic parametetr of fourth- and second-order field problems and some other inherently ill-conditioned problems.
An extreme-scale implicit solver for complex PDEs
- Rudi