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We present a spectral/hp element method for a depth-integrated Boussinsq model for the efficient simulation of nonlinear wave-body interaction. The model exploits a `unified' Boussinesq framework, i.e. the flow under the body is also treated with the depth-integrated approach, initially proposed by (Jiang, 2001) and more recently rigorously analysed by (Lannes, 2016). The choice of the Boussinesq equations allows the elimination of the vertical dimension, resulting in a wave-body model with an adequate precision for weakly nonlinear and dispersive waves expressed in horizontal dimensions only. The framework involves the coupling of two different domains with different flow characteristics. In this work we employ flux-based conditions for domain coupling, following the recipes provided by the discontinuous Galerkin spectral/hp element framework. Inside each domain, the continuous spectral/hp element method is used to solve the appropriate flow model. The spectral/hp element method allows to achieve high-order, possibly exponential, convergence for non-breaking waves and account for the nonlinear interaction with fixed and floating bodies. Our main contribution is to include floating surface-piercing bodies in the conventional depth-integrated Boussinesq framework and the use of a spectral/hp element method for high-order accurate numerical discretization in space. The model is validated against published results for wave-body interaction and confirmed to have excellent accuracy. The proposed nonlinear model is demonstrated to be relevant for the simulation of wave energy devices.

In the discontinuous Galerkin framework, a generalized anisotropic wave impedance is proposed, to succinctly solve the Riemann problem for 3-D full-anisotropic poroelastic media. Consequently, the eigenvalue problem for the large hyperbolic system of poroelastic waves is effectively simplified from the rank of 13 to 4, indicating four types of waves: two P waves due to the porosity, and two S waves due to the anisotropy. Moreover, the domain decomposition is implemented by the nonconformal-mesh technique to adaptively distribute grid sizes. In addition, the perfectly matched layer is used to truncate the finite computational domain. Verifications with an independent finite-difference code and an analytical solution illustrate the accuracy and flexibility of our algorithm.

In this article, we present a time-dependent dual-porosity–Navier–Stokes model with four interface conditions , including Beavers–Joseph interface condition, to describe a coupling system of complex porous media and conduit networks. This system has many applications, such as the flow simulation problems for a multistage fractured horizontal wellbore with suitable boundary/interface conditions and complex geometries . For this coupling system, a decoupled finite element method is proposed based on Robin type transmission conditions . In order to avoid the iteration for the traditional domain decomposition , the interface information , which is needed for the Robin type transmission conditions at the current time step, is predicted directly from the numerical solution of the previous time steps. The stability and convergence of this Robin-type decoupled finite element method are rigorously analyzed. A series of analysis techniques are utilized to analyze various components of this complicated multi-physics problem term by term, especially for the Beavers–Joseph interface condition, the interaction terms of dual-porosity model, and the nonlinear advection of Navier–Stokes equation. Moreover, a pair of Robin parameters are considered in the scheme and analysis, and their effects on the robustness for the case with low permeability and low storativity are numerically investigated. Numerical experiments are provided to validate the convergence of the proposed algorithm and illustrate the features of the application to flow problems around multistage fractured horizontal wellbore.

This paper proposes an efficient proper orthogonal decomposition with interpolation (PODI) method for real-time simulation of indenter contact problems. In the offline stage, POD models are constructed from displacement and von-Mises stress snapshots of full-order finite element solutions of indenter contact problems with training contact locations and indentation depths. Indentation depth and contact location are respectively used to interpolate POD basis vectors and POD coefficients by Lagrange interpolation with a congruence transformation and radial basis function (RBF) interpolation. In the online stage, displacements and von-Mises stress of solids under a new indentation loading are obtained very quickly by using the interpolated POD matrix and POD coefficients. A clustering-based contact domain decomposition (CDD) scheme is proposed to further improve the computational performance of the present PODI–RBF​ method. Numerical results show that the PODI–RBF method using the CDD scheme can be used to obtain finite element solutions of indenter contact problems at 400 ∼ 900 Hz on a personal computer for finite element models with 8000 ∼ 16000 nodes, which is promising to real-time simulation of indenter contact problems.

Isogeometric analysis (IGA) is a novel computer aided engineering technique that addresses diverse problems in computational mechanics [ [1] , [2] , [3] , [4] ], all under the exact geometric representation. Apart from the exact geometric representation, the high continuity of IGA shape functions enhances the accuracy and robustness of the method. However, the price to pay is that the resulting matrices are denser, with increased bandwidth and overlapping which make the solution of large-scale problems more computationally intensive. As a result, effective solution techniques are still considered an open issue for further research. In this paper, an innovative family of solution schemes based on domain decomposition methods (DDM) is proposed, that significantly reduces the computational cost. Specifically, the solution of the global system is performed with the preconditioned conjugate gradient algorithm (PCG) whose preconditioning step is evaluated with a dual DDM where special care is taken to avoid the overlapped subdomains which are inherent in decomposed IGA formulation due to the increased continuity of the shape functions.

An efficient and accurate method is proposed to solve the incompressible flow momentum and continuity equations in computational domains partitioned into subdomains in the framework of the smoothed particle hydrodynamics method. The procedure does not require any overlap of the subdomains, which would result in the increase of the computational effort. Perfectly matching solutions are obtained at the surfaces separating neighboring blocks. The block interfaces can be both planar and curved surfaces allowing to easily decompose even geometrically complex domains. The smoothing length of the kernel function is maintained constant in each subdomain, while changing between blocks where a different resolution is required. Particles leaving each block through the interfaces are deactivated and correspondingly new particles are generated at the neighboring block using a dynamically adaptive procedure to control their frequency of release. No splitting and coalescing method is thus employed to take into account the different size and mass of the particles going through the interfaces. Mass conservation is guaranteed during the procedure, which is a challenging task in a Lagrangian method based on the domain decomposition. The test cases in both 2D and 3D approximation show the accuracy of the method and its ability to strongly reduce the computational efforts through a multi-resolution approach.

Stochastic spectral finite element models of practical engineering systems may involve solutions of linear systems or linearized systems for non-linear problems with billions of unknowns. For stochastic modeling, it is therefore essential to design robust, parallel and scalable algorithms that can efficiently utilize high-performance computing to tackle such large-scale systems. Domain decomposition based iterative solvers can handle such systems. Although these algorithms exhibit excellent scalabilities, significant algorithmic and implementational challenges exist to extend them to solve extreme-scale stochastic systems using emerging computing platforms. Intrusive polynomial chaos expansion based domain decomposition algorithms are extended here to concurrently handle high resolution in both spatial and stochastic domains using an in-house implementation. Sparse iterative solvers with efficient preconditioners are employed to solve the resulting global and subdomain level local systems through multi-level iterative solvers. Parallel sparse matrix–vector operations are used to reduce the floating-point operations and memory requirements. Numerical and parallel scalabilities of these algorithms are presented for the diffusion equation having spatially varying diffusion coefficient modeled by a non-Gaussian stochastic process. Scalability of the solvers with respect to the number of random variables is also investigated.

The Richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated/unsaturated flow in porous media. We assume that the medium occupies a bounded Lipschitz domain partitioned into two disjoint subdomains separated by a fixed interface $\Gamma$. This leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at $\Gamma$. After an Euler implicit discretisation of the resulting nonlinear subproblems a linear iterative ($L$-type) domain decomposition scheme is proposed. The convergence of the scheme is proved rigorously. In the last part we present numerical results that are in line with the theoretical finding, in particular the unconditional convergence of the scheme. We further compare the scheme to other approaches not making use of a domain decomposition. Namely, we compare to a Newton and a Picard scheme. We show that the proposed scheme is more stable than the Newton scheme while remaining comparable in computational time, even if no parallelisation is being adopted. Finally we present a parametric study that can be used to optimize the proposed scheme., Comment: 34 pages, 13 figures, 7 tables

We present a neural network-based method for solving linear and nonlinear partial differential equations, by combining the ideas of extreme learning machines (ELM), domain decomposition and local neural networks. The field solution on each sub-domain is represented by a local feed-forward neural network, and $C^k$ continuity is imposed on the sub-domain boundaries. Each local neural network consists of a small number of hidden layers, while its last hidden layer can be wide. The weight/bias coefficients in all hidden layers of the local neural networks are pre-set to random values and are fixed, and only the weight coefficients in the output layers are training parameters. The overall neural network is trained by a linear or nonlinear least squares computation, not by the back-propagation type algorithms. We introduce a block time-marching scheme together with the presented method for long-time dynamic simulations. The current method exhibits a clear sense of convergence with respect to the degrees of freedom in the neural network. Its numerical errors typically decrease exponentially or nearly exponentially as the number of degrees of freedom increases. Extensive numerical experiments have been performed to demonstrate the computational performance of the presented method. We compare the current method with the deep Galerkin method (DGM) and the physics-informed neural network (PINN) in terms of the accuracy and computational cost. The current method exhibits a clear superiority, with its numerical errors and network training time considerably smaller (typically by orders of magnitude) than those of DGM and PINN. We also compare the current method with the classical finite element method (FEM). The computational performance of the current method is on par with, and oftentimes exceeds, the FEM performance., Comment: 62 pages, 48 figures, 12 tables

In intrusive methods for the stochastic analysis of engineering systems, the solution of the stochastic partial differential equations leads to an augmented algebraic system of equations, with respect to the corresponding deterministic problem. The solution of this augmented system can become quite challenging or even impossible due to the increased computational resources and effort required, especially in large-scale problems with large parameter variability. Recent developments in computer hardware regarding graphics processing units (GPU)-accelerated computing have been proven very promising due to their advanced capabilities and have been used in a number of scientific fields in order to enhance the solution of computationally high demanding problems. In this work, the benefits achieved with the exploitation of the GPU capabilities are demonstrated, in addressing intrusive stochastic mechanics problems where the solution of the resulting finite element algebraic equations is performed with the dual domain decomposition method, implementing specifically tailored preconditioners. The results show a significant enhancement of the spectral stochastic finite element method in terms of computational performance as well as of the consumed energy efficiency when applying the proposed methodology.

A novel domain-decomposition least-squares Petrov-Galerkin (DD-LSPG) model-reduction method applicable to parameterized systems of nonlinear algebraic equations (e.g., arising from discretizing a parameterized partial-differential-equations problem) is proposed. In contrast with previous works, we adopt an algebraically non-overlapping decomposition strategy rather than a spatial-decomposition strategy, which facilitates application to different spatial-discretization schemes. Rather than constructing a low-dimensional subspace for the entire state space in a monolithic fashion, the methodology constructs separate subspaces for the different subdomains/components characterizing the original model. In the offline stage, the method constructs low-dimensional bases for the interior and interface of components. In the online stage, the approach constructs an LSPG ROM for each component and enforces strong or weak compatibility on the 'ports' connecting them. We propose four different ways to construct reduced bases on the interface/ports of subdomains and several ways to enforce compatibility across connecting ports. We derive a posteriori and a priori error bounds for the DD-LSPG solutions. Numerical results performed on nonlinear benchmark problems in heat transfer and fluid dynamics demonstrate that the proposed method performs well in terms of both accuracy and computational cost, with different choices of basis and compatibility constraints yielding different performance profiles., 43 pages, 17 figures

A novel mathematical model for fiber-reinforced materials is proposed. It is based on a 1-dimensional beam model for the thin fiber structures, a flexible and general 3-dimensional elasticity model for the matrix and an overlapping domain decomposition approach. From a computational point of view, this is motivated by the fact that matrix and fibers can be easily meshed independently. Our main interest is in fiber-reinforced polymers where the Young modulus is quite different. Thus the modeling error from the overlapping approach is of no significance. The coupling conditions acknowledge both, the forces and the moments of the beam model and transfer them to the background material. A suitable static condensation procedure is applied to remove the beam balance equations. The condensed system then forms our starting point for a numerical approximation in terms of isogeometric analysis. The choice of our discrete basis functions of higher regularity is motivated by the fact, that as a result of the static condensation, we obtain second gradient terms in fiber direction. Eventually, a series of benchmark tests demonstrate the flexibility and robustness of the proposed methodology. As a proof-of-concept, we show that our new model is able to capture bending, torsion and shear dominated situations.

This paper aims at developing an efficient preconditioned iterative domain decomposition (DD) method for the sampling of linear stochastic elliptic equations . To this end, we consider a non-overlapping DD method resulting in a Symmetric Positive Definite (SPD) Schur system for almost every sampled problem. To accelerate the iterative solution of the Schur system, we propose a new stochastic preconditioning strategy that produces a preconditioner adapted to each sampled problem and converges toward the ideal preconditioner ( i.e. , the Schur operator itself) when the numerical parameters increase. The construction of the stochastic preconditioner is trivially parallel and takes place in an off-line stage, while the evaluation of the sample’s preconditioner during the sampling stage has a low and fixed cost. One key feature of the proposed construction is a factorized form combined with Polynomial Chaos expansions of local operators. The factorized form guarantees the SPD character of the sampled preconditioners while the local character of the PC expansions ensures a low computational complexity. The stochastic preconditioner is tested on a model problem in 2 space dimensions. In these tests, the preconditioner is very robust and significantly more efficient than the deterministic median-based preconditioner, requiring, on average, up to 7 times fewer iterations to converge. Complexity analysis suggests the scalability of the preconditioner with the number of subdomains.

We are interested in a reduced order method for the efficient simulation of blood flow in arteries. The blood dynamics is modeled by means of the incompressible Navier–Stokes equations. Our algorithm is based on an approximated domain-decomposition of the target geometry into a number of subdomains obtained from the parametrized deformation of geometrical building blocks (e.g., straight tubes and model bifurcations). On each of these building blocks, we build a set of spectral functions by Proper Orthogonal Decomposition of a large number of snapshots of finite element solutions (offline phase). The global solution of the Navier–Stokes equations on a target geometry is then found by coupling linear combinations of these local basis functions by means of spectral Lagrange multipliers (online phase). Being that the number of reduced degrees of freedom is considerably smaller than their finite element counterpart, this approach allows us to significantly decrease the size of the linear system to be solved in each iteration of the Newton–Raphson algorithm. We achieve large speedups with respect to the full order simulation (in our numerical experiments, the gain is at least of one order of magnitude and grows inversely with respect to the reduced basis size), whilst still retaining satisfactory accuracy for most cardiovascular simulations.

Metals subjected to high strain rate impact often exhibit a sudden and profound drop in the material’s load bearing capability, a ductile failure phenomenon known as shear banding. Shear bands, characterized as material instabilities, are driven by shear heating and described as narrow regions that have sustained intense plastic deformation and high temperature rise. This coupled thermo-mechanical localization problem can be formulated as a nonlinear system with two balance equations, momentum and energy, and two constitutive laws for elasticity and plasticity. In our formulation, mixed finite elements are used to discretize the equations in space and an implicit finite difference scheme is used to advance the system in time, where at every time step, a Newton type method is used to solve the system monolithically. To that end, a block Jacobian matrix is formed analytically using Gâteaux derivatives. The resulting Jacobian is sparse, nonsymmetric and its sparsity pattern and eigenvalue content vary with the different stages of shear bands formation, consequently posing a significant challenge to iterative solvers. To address this issue, an overlapping Domain Decomposition preconditioner that takes into account the physics of shear bands and the domain in which it forms, is proposed. The key idea is to resolve the shear band domain accurately while only solving the remaining domain approximately, with selective updates when necessary. The preconditioner is formed on the basis of an additive Schwartz method that is applied to a Schur complement partition of the system. The proposed preconditioner is implemented in parallel and tested on three different benchmark problems as compared against off-the-shelf solvers. We study the effect of hh- and kk-refinement that are obtained from Isogeometric discretizations of the system, the overlapping strategy and its parallel performance. Excellent performance is demonstrated in serial and parallel on all benchmark examples.

This paper presents a simple yet flexible method for coupling non-conforming NURBS patches in isogeometric frameworks. It consists of virtually refining bordering patches to derive coupling constraints in a master-slave formulation. These constraints are then enforced via a substitution method for condensation of the slave variables, thereby reducing the model size. In the special case of hierarchical meshes, the method results in an exact connection. For an arbitrary non-conforming configuration, the master-slave formulation takes the interface constraints into account in a least-squares sense. This yields an accurate weak coupling without overconstraining the interface. The coupling relationships only depend on the mesh itself and not on any problem-dependent parameters, allowing them to be generated in a pre-processing step with very limited numerical efforts. Besides this low computational cost, the main merit of the proposed method lies in its simplicity and robustness, yielding good results for arbitrarily strongly non-conforming patch configurations. Several numerical examples are studied for different problem types requiring C0- or C1-continuity, in particular time-harmonic acoustics and (dynamic) thin plate bending. Benchmarking of the proposed approach against existing similar techniques illustrates its superior accuracy and robustness. publisher: Elsevier articletitle: A robust patch coupling method for NURBS-based isogeometric analysis of non-conforming multipatch surfaces journaltitle: Computer Methods in Applied Mechanics and Engineering articlelink: http://dx.doi.org/10.1016/j.cma.2016.06.022 content_type: article copyright: © 2016 Elsevier B.V. All rights reserved. ispartof: Computer Methods in Applied Mechanics and Engineering vol:316 pages:235-260 status: published

In this paper we consider a new version of the dual-primal isogeometric tearing and interconnecting (IETI-DP) method for solving large-scale linear systems of algebraic equations arising from discontinuous Galerkin (dG) isogeometric analysis of diffusion problems on multipatch domains with non-matching meshes. The dG formulation is used to couple the local problems across patch interfaces. The purpose of this paper is to present this new method and provide numerical examples indicating a polylogarithmic condition number bound for the preconditioned system and showing an incredible robustness with respect to large jumps in the diffusion coefficient across the interfaces., Comment: arXiv admin note: text overlap with arXiv:1511.07183

In this paper, a parallel domain decomposition method is proposed, for solving the fully-mixed Stokes–Darcy coupled problem with the Beavers–Joseph–Saffman (BJS) interface conditions. With newly constructed Robin-type boundary conditions, the present method adopts modified weak formulation to decouple the original problem into two independent subproblems. The equivalence between the original problem and the decoupled subproblems is derived under some compatibility conditions. Another equivalence of two weak formulations with different spaces is also established, for subsequent convergence analysis based on the decoupled modified weak formulation. Moreover, the convergence of the iterative parallel method in a more general framework is shown. With some suitable choice of parameters, both mesh-dependent and mesh-independent convergence rates are proved rigorously. Finally, we present several numerical examples to show the exclusive features of the proposed method.

We discuss two linearization and domain decomposition methods for mathematical models for two-phase flow in a porous medium. The medium consists of two adjacent regions with possibly different parameterizations. The model accounts for non-equilibrium effects like dynamic capillarity and hysteresis. The θ -scheme is adopted for the temporal discretization of the equations yielding nonlinear time-discrete equations. For these, we propose and analyze two iterative schemes, which combine a stabilized linearization iteration of fixed-point type, the L-scheme, and a non-overlapping domain decomposition method into one iteration. First, we prove the existence of unique solutions to the problems defining the linear iterations. Then, we give the rigorous convergence proof for both iterative schemes towards the solution of the time-discrete equations. The developed schemes are independent of the spatial discretization or the mesh and avoid the use of derivatives as in Newton based iterations. Their convergence holds independently of the initial guess, and under mild constraints on the time step. The numerical examples confirm the theoretical results and demonstrate the robustness of the schemes. In particular, the second scheme is well suited for models incorporating hysteresis. The schemes can be easily implemented for realistic applications.

The aim of this work is to construct and analyze a FETI-DP type domain decomposition preconditioner for isogeometric discretizations of the Stokes and mixed linear elasticity systems. This method extends to the isogeometric analysis context the preconditioner previously proposed by Tu and Li (2015) for finite element discretizations of the Stokes system. The resulting isogeometric FETI-DP algorithm is proven to be scalable in the number of subdomains and has a quasi-optimal convergence rate bound which is polylogarithmic in the ratio of subdomain and element sizes. Extensive two-dimensional numerical experiments validate the theory, investigate the behavior of the preconditioner with respect to both the spline polynomial degree and regularity, and show its robustness with respect to domain deformation, material incompressibility and presence of elastic coefficient discontinuities across subdomain interfaces.

To tackle general sub-domain problems in geomechanics, we present an MFEM scheme on curved interfaces based on NURBS curves and surfaces. The goal is to have a more robust geometrical representation for mortar spaces, which allows gluing non-conforming interfaces on realistic geometries. The resulting mortar saddle-point problem is decoupled using standard domain decomposition techniques such as Dirichlet–Neumann, to exploit current parallel machine architectures. Two- and three-dimensional examples ranging from near-wellbore applications to field level subsidence computations show that the proposed scheme can handle problems of practical interest.

International audience; A non-overlapping domain decomposition method is proposed to solve large-scale finite element models for the propagation of sound with a background mean flow. An additive Schwarz algorithm is used to split the computational domain into a collection of sub-domains, and an iterative solution procedure is formulated in terms of unknowns defined on the interfaces between sub-domains. This approach allows to solve large-scale problems in parallel with only a fraction of the memory requirements compared to the standard approach which is to use a direct solver for the complete problem. While domain decomposition techniques have been used extensively for Helmholtz problems, this is the first application to aero-acoustics. The optimized Schwarz formulation is extended to the linearized potential theory for sound waves propagating in a potential base flow. A high-order finite element method is used to solve the governing equations in each sub-domain, and well-designed interface conditions based on local approximations of the Dirichlet-to-Neumann map are used to accelerate the convergence of the iterative procedure. The method is assessed on an academic test case and its benefit demonstrated on a realistic turbofan engine intake configuration.

In this paper, an efficient numerical procedure is proposed, which is adapted to the modal analysis of a fully-loaded spent nuclear fuel canister exhibiting distinct structural levels associated with a hierarchy of components. On the one hand, the fully-loaded spent nuclear fuel canister is constituted of a repetition of identical components, resulting in a pseudo-periodicity; and the components of a given level are separated from each other and only connected to their upper level through localized attachments. This gives rise to an advantageous structural connectivity that can be exploited for efficiency. On the other hand, the necessary fine mesh resolution of the small levels leads to a high-dimensional computational model and, in addition, the independent resonant vibrations of each of the components produce a very large number of vibration eigenmodes. The aforementioned opportunities and difficulties are respectively leveraged and tackled by an adapted method that combines domain decomposition, shift–invert Lanczos eigenvalue solver, and Craig–Bampton substructuring technique. A parallel eigensolution via spectrum slicing is facilitated by an efficient block factorization by Schur complement due to the sparsity of the Craig–Bampton matrices. A computational gain of four orders of magnitude is obtained, at the expense of negligible errors that are exactly characterized. The proposed methodology enables a high-fidelity vibration analysis of the sealed fully-loaded spent nuclear fuel canister, useful for non-intrusive inverse identification of the structural integrity of the internal structural levels holding the nuclear material.

In this paper we present a finite-element based reduced order model and, in particular, we consider two aspects related to the introduction of inter-element boundary terms in the formulation. The first is a domain decomposition strategy in which the transmission conditions involve boundary terms to account for non-matching meshes and discontinuous physical properties. The second is a coarse mesh hyper-reduction for which we propose an adaptive refinement driven by an a posteriori error estimator that contains element boundary terms. As the finite element full order model, the reduced order model is based on the Variational Multi-Scale framework, with sub-grid scales defined not only in the element interiors, but also on the inter-element boundaries. We present some examples of application using the incompressible Navier–Stokes equations and the Boussinesq approximation. R. Reyes acknowledges the scholarship received from COLCIENCIAS, from the Colombian Government. R. Codina acknowledges the support received from the ICREA, Spain Acadèmia Research Program, from the Catalan Government.

This paper presents a nonlinear multiscale approach for periodic structures in the quasi-static, small strain regime. The approach consists in combining a domain decomposition method in which interface conditions are established through a “fictitious” frame with reduced-order modeling (ROM). We propose to approximate interface displacements, subdomain displacements and Lagrange Multipliers as linear combinations of reduced sets of dominant modes, and to assign to the coefficients of such linear combinations the role of coarse-scale displacements, strains and stresses, respectively. We discuss and propose a method based on subspace rotations that ensures that these three modal approximations lead to stable formulations. The modes of such expansions are determined in an offline “training” stage by applying the truncated Singular Value Decomposition (SVD) to the solutions obtained from Finite Element (FE) analyses. In contrast to other multiscale approaches, such as FE-ROM homogenization schemes, which uses a single unit cell with periodic conditions, here the “training” structures are formed by several unit cells. This way, we allow the SVD to extract also dominant patterns of interaction between subdomains in terms of reactive forces and deformations. This original feature confers to the proposed approach three unique advantages over FE-ROM homogenization schemes, namely: (1) It can deal with unit cells of arbitrary size (no need for scale separation). (2) It can model, not only how forces are transmitted through the structure, but also local effects. (3) It can also handle domains which are only periodic along one or two directions (beam-like or shell structures, respectively). To deal with material nonlinearities, element-wise Gauss integration of reduced internal forces is replaced by an algorithmically improved version of the hyperreduction scheme recently proposed by the author elsewhere, called the Empirical Cubature Method(ECM). Furthermore, we demonstrate that the coarse-scale Degrees of Freedom (DOFs) of a subdomain can be expressed directly in terms of the (fine-scale) stresses and strains at the ECM integration points. This feature dispenses with the need of deploying a special algorithmic infrastructure of intertwined local/global problems, as it occurs in FE-ROM schemes, since the unit cell can be treated as a special type of finite element, in which the centroids of the interfaces and the ECM points play the role of nodes and Gauss points, respectively. To illustrate all these advantages, we present 4 distinct examples: two beam-like structures of rectangular-shape and I-shaped cross-sections, a 2D hexagonal cellular material, and a cylindrical shell made of a porous composite cell. In all 4 cases the proposed method is able to produce coarse-scale models reducing the number of DOFs and integration points by over two orders of magnitude, with errors below 5 %. Interestingly, in the case of the beam-like structures, we show that the method provides 2-node beam finite elements whose kinematics are identical to that predicted by “analytical” beam theories (6 DOFs per node in the case of the rectangular beam), and totally consistent with their 3D full-order counterparts. The research leading to these results has received funding from ANACONDA BIOMED S.L. and the Spanish Ministry of Science, Innovation and Universities via grant RTC-2017-6749-1. The author also acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2018-000797-S).

International audience; A non-overlapping domain decomposition method (DDM) is proposed for the parallel finite-element solution of large-scale time-harmonic wave problems. It is well-known that the convergence rate of this kind of method strongly depends on the transmission condition enforced on the interfaces between the subdomains. Local conditions based on high-order absorbing boundary conditions (HABCs) have proved to be well-suited, as a good compromise between basic impedance conditions, which lead to suboptimal convergence, and conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain — which are too expensive to compute. However, a direct application of this approach for configurations with interior cross-points (where more than two subdomains meet) and boundary cross-points (points that belong to both the exterior boundary and at least two subdomains) is suboptimal and, in some cases, can lead to incorrect results.In this work, we extend a non-overlapping DDM with HABC-based transmission conditions approach to efficiently deal with cross-points for lattice-type partitioning. We address the question of the cross-point treatment when the HABC operator is used in the transmission condition, or when it is used in the exterior boundary condition, or both. The proposed cross-point treatment relies on corner conditions developed for Padé-type HABCs. Two-dimensional numerical results with a nodal finite-element discretization are proposed to validate the approach, including convergence studies with respect to the frequency, the mesh size and the number of subdomains. These results demonstrate the efficiency of the cross-point treatment for settings with regular partitions and homogeneous media. Numerical experiments with distorted partitions and smoothly varying heterogeneous media show the robustness of this treatment.

We propose a conservative physics-informed neural network (cPINN) on discrete domains for nonlinear conservation laws. Here, the term discrete domain represents the discrete sub-domains obtained after division of the computational domain, where PINN is applied and the conservation property of cPINN is obtained by enforcing the flux continuity in the strong form along the sub-domain interfaces. In case of hyperbolic conservation laws, the convective flux contributes at the interfaces, whereas in case of viscous conservation laws, both convective and diffusive fluxes contribute. Apart from the flux continuity condition, an average solution (given by two different neural networks) is also enforced at the common interface between two sub-domains. One can also employ a deep neural network in the domain, where the solution may have complex structure, whereas a shallow neural network can be used in the sub-domains with relatively simple and smooth solutions. Another advantage of the proposed method is the additional freedom it gives in terms of the choice of optimization algorithm and the various training parameters like residual points, activation function, width and depth of the network etc. Various forms of errors involved in cPINN such as optimization, generalization and approximation errors and their sources are discussed briefly. In cPINN, locally adaptive activation functions are used, hence training the model faster compared to its fixed counterparts. Both, forward and inverse problems are solved using the proposed method. Various test cases ranging from scalar nonlinear conservation laws like Burgers, Korteweg–de Vries (KdV) equations to systems of conservation laws, like compressible Euler equations are solved. The lid-driven cavity test case governed by incompressible Navier–Stokes equation is also solved and the results are compared against a benchmark solution. The proposed method enjoys the property of domain decomposition with separate neural networks in each sub-domain, and it efficiently lends itself to parallelized computation, where each sub-domain can be assigned to a different computational node.

In this paper we present a high-order contact element using a mortar-based domain decomposition method. The main objective is to verify the solution accuracy by increasing the element interpolation order in two-dimensional contact problems, considering large deformations and friction. We present studies of accuracy, processing time and contact stress oscillation using h - and p -refinements. The comparative results show that the high-order interpolation is a strategy with superior performance for the contact problems analysed. The results for the p -refinement improved the solution accuracy of the stresses and forces generated by contact with a lower processing time.

We consider a second order elliptic problem with a heterogeneous coefficient, which models, for example, single phase flow through a porous medium. We write this problem in mixed form and approximate it for parallel computation using the multiscale mortar domain decomposition mixed finite element method, which gives rise to a saddle point linear system. We use a relatively fine mortar space, which allows us to enforce continuity of the normal velocity flux, or nearly so in the case of nonmatching meshes. To solve the Schur complement linear system for the mortar unknowns, we propose a two-level preconditioner based on the interfaces between subdomains. The coarse preconditioner also uses the multiscale mortar domain decomposition method, but with instead a very coarse mortar space. We show that the prolongation operator of the coarse mortar to the fine is defined uniquely by the condition that the L 2 -projection of a coarse mortar agrees with its projection onto the space of normal velocity fluxes, i.e., no energy is introduced when changing mortar scales. The local smoothing preconditioner is based on block Jacobi, using blocks defined by the interfaces. We use restrictive smoothing domains that are smaller normal to the interfaces, and overlapping in the directions tangential to the interfaces. In the simplest case, the condition number of the preconditioned interface operator is bounded by a multiple of ( log ( 1 + H / h ) ) 2 . We show several numerical examples involving strongly heterogeneous porous media to demonstrate the efficiency and robustness of the preconditioner. We see that it is often desirable, and sometimes necessary, to use a piecewise linear or higher order coarse mortar space to achieve good convergence for heterogeneous problems.

This paper proposes a method for decomposing two-dimensional domains into subdomains for parameterization and isogeometric analysis. Given a complex domain in a plane, the skeleton of the domain is computed to guide the domain decomposition. A continuous parameterization of the domain is then obtained by parameterizing each respective subdomain. This parameterization method is applied with isogeometric analysis to solve numerical PDEs over two-dimensional domains. Examples are provided to demonstrate that the new parameterization method is superior to other state-of-the-art parameterization techniques and that it performs better in isogeometric analysis.

In this paper we introduce the idea of combining artificial compressibility (AC) with quasi-Newton (QN) methods to solve strongly coupled, fully/quasi-enclosed fluid–structure interaction (FSI) problems. Partitioned, incompressible, FSI based on Dirichlet–Neumann domain decomposition solution schemes cannot be applied to problems where the fluid domain is fully enclosed. A simple example often provided in literature is that of a balloon with a prescribed inflow velocity. In this context, artificial compressibility (AC) is a useful method by which the incompressibility constraint can be relaxed by including a source term within the fluid continuity equation. The attractiveness of AC stems from the fact that this source term can readily be added to almost any fluid field solver, including most commercial solvers. Once included, both the modified fluid solver and structural solver can be treated as “black-box” field operators. AC is however limited to the class of problems it can effectively be applied to. For example, AC is an efficient solution strategy for the simulation of blood flow through arteries, but performs poorly when applied to the simulation of blood flow through an opening heart valve. The focus of this paper is thus to extend the application of AC by including an additional Newton system accounting for the missing interface sensitivities. We do so through the use of a multi-vector update quasi-Newton (MVQN) method, where the required system Jacobians are approximated rather than explicitly computed. In so doing, we continue to facilitate the notion that the AC modified fluid field solver and solid field solver can be treated as “black-box” solvers. We aim to demonstrate the improved performance of the combination of AC+QN when compared to AC applied in isolation.

This article suggests a new and efficient method, called Immersed Volume Method (IVM) to compute compressible viscous flows with complex stationary geometries using finite difference codes with a staggered cartesian grid formulation. A background cartesian mesh is generated for each staggered variable and a finite volume approach is adopted in the layer near the immersed boundaries according to the well known cut cell method. Accurate description of the real three-dimensional geometry inside the cut cell volume is preserved by means of a triangulated surface description instead of approximating it by a plane. The overall spatial accuracy of the base solver (2nd order in this article) is preserved by means of high order flux reconstruction in the cut cells. In particular, the code adopted here to test the suggested technique adopts a 2nd order centered scheme in space and 3rd order Runge–Kutta scheme in time. The code is parallelized using domain decomposition and message passing interface. The robustness and accuracy of the method is proved simulating, at first, a laminar flow past a sphere at R e = 50 , 200 , 250 , then a turbulent nonreacting flow past a sphere with a sting at R e = 51 500 and a turbulent premixed, stoichiometric CH 4 / air flame anchored by means of a cubic bluff body at R e = 3200 , both adopting the Large Eddy Simulation approach. Numerical results are compared with available experimental data.

Isogeometric collocation methods are very recent and promising numerical schemes that preserve the advantages of isogeometric analysis but often exhibit better performances than their Galerkin counterparts. In the present paper, an additive overlapping Schwarz method for isogeometric collocation discretizations is introduced and studied. The resulting preconditioner, accelerated by GMRES, is shown to be scalable with respect to the number of subdomains and very robust with respect to the isogeometric discretization parameters such as the mesh size and polynomial degree, as well as with respect to the presence of discontinuous elliptic coefficients and domain deformations.

A non-intrusive global/local algorithm with non-matching (incompatible) interface is proposed to evaluate the durability of large-scale structures with local nonlinearity both on the basis of local multigrid strategy and the domain decomposition (DD) method, which can serve as an intermediate way to perform analyses of complex structures. The proposed algorithm couples a global linear elastic analysis (on coarse mesh) with a local nonlinear analysis (on fine mesh), which extends the iterative global/local method to treat non-matching partition interfaces. Furthermore, to deal with non-matching interface the framework of localized Lagrange multiplier (LLM) method is introduced, and a feasible but robust data transfer method is established with radial basis function (RBF) interpolation. Aitken’s acceleration method is adopted to increase the convergence rate. The convergence property and accuracy of this non-intrusive algorithm are investigated for a two-dimensional model with different interface geometries and interface node allocations. The effect of reduced integration on the convergence rate is also concerned in verification. Then, it is further verified with a three-dimensional model. Numerical results reveal that this non-intrusive algorithm is robust in the non-matching interface data transfer procedure, which is significant in error control during iterations, and it has good performance in both convergence and accuracy.

A non-overlapping domain decomposition method is presented to solve a coupled Stokes–Darcy flow problem in parallel by partitioning the computational domain into multiple subdomains, upon which families of coupled local problems of lower complexity are formulated. The coupling is based on appropriate interface matching conditions. The global problem is reduced to an interface problem by eliminating the interior subdomain variables. The interface problem is solved by an iterative procedure, which requires solving subdomain problems at each iteration. Finite element techniques appropriate for the type of each subdomain problem are used to discretize it. The condition number of the resulting algebraic system is analyzed and numerical tests verifying the theoretical estimates are provided.

The spectral stochastic finite element method (SSFEM) may offer an efficient alternative to the traditional Monte Carlo simulations (MCS) for uncertainty quantification of large-scale numerical simulations. In the framework of the intrusive SSFEM, the main computational challenge involves solving a coupled set of deterministic linear systems. For large-scale numerical models, the computational efficiency of the intrusive SSFEM primarily depends on the solution techniques employed to tackle the resulting coupled linear systems. In this paper, we report a probabilistic version of the dual-primal domain decomposition method for the intrusive SSFEM in order to exploit high performance computing platforms for uncertainty quantification. In particular, we formulate a probabilistic version of the dual-primal finite element tearing and interconnect (FETI-DP) technique to solve the large-scale linear systems in the intrusive SSFEM. In the probabilistic setting, the operator of the dual interface system in the dual-primal approach contains a coarse problem. The introduction of the coarse problem in the probabilistic setting leads to a scalable performance of the dual-primal iterative substructuring method for uncertainty quantification of large-scale computational models. The convergence properties, numerical and parallel scalabilities of the probabilistic FETI-DP method and the recently developed probabilistic version of the balancing domain decomposition by constraints (BDDC) method are contrasted. For numerical illustrations, we consider flow through porous media and linear elasticity problems with spatially varying system parameters modelled as non-Gaussian random processes. The algorithms are implemented on a Linux cluster using MPI and PETSc parallel libraries.

In this paper, a novel domain decomposition method is introduced to solve a large finite element model. In this method, the decomposed domains do not overlap and are connected using a simple connective finite element, which influences the nodal point equilibrium between adjacent finite elements. This approach has the advantage that it allows use of a direct method such as Gauss elimination even in a singular problem. The singular stiffness matrices from the floating domain without the Dirichlet boundary conditions are changed into invertible stiffness matrices by assembling the connective elements. Another advantage is that computational time and storage can be reduced by using a banded matrix in the direct solver. In order to describe this proposed method, we first review the FETI method, which is the most popular domain decomposition method. Then the proposed method is introduced with a technical approach in a distributed computer system. Finally, the high scalability and computational efficiency of the proposed method are verified by comparing with the traditional FETI method for 2D and 3D finite element examples which have floating subdomains. In the result, we demonstrate high scalability for a large finite element model.

We extend the static condensation reduced basis element (scRBE) method to treat the class of parametrized complex Helmholtz partial differential equations. The main ingredients are (i) static condensation at the interdomain level, (ii) a conforming eigenfunction “port” representation at the interface level, (iii) the reduced basis (RB) approximation of finite element (FE) bubble functions at the intradomain level, and (iv) rigorous system-level error bounds which reflect RB perturbation of the FE Schur complement. We then incorporate these ingredients in an Offline–Online computational strategy to achieve rapid and accurate prediction of parametric systems formed from instantiations of interoperable parametrized archetype components from a Library. We introduce a number of extensions with respect to the original scRBE framework: first, primal–dual RB methods for general non-symmetric (complex) problems; second, stability constant procedures for weakly coercive problems (at both the interdomain level and intradomain level); third, treatment of non-port linear–functional outputs (as well as functions of outputs); fourth, consideration of particular components and outputs relevant to acoustic applications. We consider several numerical examples in acoustics (in particular focused on mufflers and horns) to demonstrate that the approach can handle models with many parameters and/or topology variations with particular reference to waveguide-like applications.

Geophysical flows over complex domains often encompass both coarse and highly resolved regions. Approximating these flows using shock-capturing methods with explicit timestepping gives rise to a Courant–Friedrichs–Lewy (CFL) timestep constraint. This approach can result in small global timesteps often dictated by flows in small regions, vastly increasing computational effort over the whole domain. One approach for coping with this problem is to use locally varying timesteps. In previous work, we formulated a local timestepping (LTS) method within a Runge–Kutta discontinuous Galerkin framework and demonstrated the accuracy and efficiency of this method on serial machines for relatively small-scale shallow water applications. For more realistic models involving large domains and highly complex physics, the LTS method must be parallelized for multi-core parallel computers. Furthermore, additional physics such as strong wind forcing can effect the choice of local timesteps. In this paper, we describe a parallel LTS method, parallelized using domain decomposition and MPI. We demonstrate the method on tidal flows and hurricane storm surge applications in the coastal regions of the Western North Atlantic Ocean.

We here propose a multiscale numerical method for the solution of stochastic parametric partial differential equations with localized uncertainties described with a finite number of random variables. It is based on a multiscale domain decomposition method that exploits the localized side of uncertainties and incidentally improves the conditioning of the problem by operating a separation of scales. An efficient iterative algorithm is proposed that requires the solution of a sequence of simple global problems at a macro scale, involving a deterministic operator, and local problems at a micro scale for which we have the possibility to use fine approximation spaces. Global and local problems are solved using tensor approximation methods allowing the representation of high dimensional stochastic parametric solutions. Convergence properties of these tensor based methods, which are closely related to spectral decompositions, benefit from the separation of scales. Different types of uncertainties are considered at the micro level. They may be associated with some variability in the operator or source terms, or even with some geometrical variability. In the latter case, specific reformulations of local problems using fictitious domain methods are introduced.

This paper presents a modification of the well established strong discontinuity approach to model failure phenomena in solids by extending it to multiple levels. This is achieved by the resolution of the overall problem to be solved into a main boundary value problem and identified sub-domains based on the concepts of domain decomposition. The initiation of those sub-domains is based on the detection of failure onset within finite elements of the main boundary value problem which takes place at the process zone in front of the propagating cracks. Those sub-domains are subsequently adaptively discretized during run-time and comprise the so called sub-boundary value problem to be solved simultaneously with the main boundary value problem. To model failure, only the sub-elements of those sub-boundary value problems are treated by the strong discontinuity approach which, depending on their state of stress, may develop strong discontinuities to be understood as jumps in the displacement field to model cracks and shear bands. Due to its resolution into many sub-elements, the single finite element of the main boundary value problem can therefore simulate a single propagating strong discontinuity arising in quasi-static problems as well as the propagation of multiple propagating strong discontinuities arising for simulations of crack branching in brittle materials undergoing dynamic failure. Whereas the advantages of the strong discontinuity approach in the form of its efficiency by statically condensing out the degrees of freedom related to the failure zone as well as its applicability to use standard displacement based, mixed, and enhanced formulations for the underlying finite element are kept, new challenges arise due to its proposed modification. Firstly, the solutions of the different sub-boundary value problems must be transferred to the main boundary value problem, which is achieved in this work based on concepts of domain decomposition. Secondly, since multiple strong discontinuities might propagate over the boundaries of the sub-boundary value problem, the applied boundary conditions must take into account the appearance of possible jumps in the displacement fields arising from the solution of the sub-boundary value problem itself. It is shown that for single propagating cracks arising in problems of quasi-static failure only minor differences are obtained through the proposed modification. For the simulation of solids undergoing dynamic fracture the modification allows though to predict the onset of crack branching without the need for any artificial crack branching criterion. A close agreement with experiments of the simulation results in terms of micro- and macro branching in addition to studying certain key parameters like critical velocity, dynamic stress intensity factor, and the strain energy release rate at branching is found.

The FETI based domain decomposition method is adapted to implement the time step of the Newmark scheme for the solution of dynamic contact problems without friction. If the ratio of the decomposition and discretization parameters is kept uniformly bounded, then the cost of the time step is proved to be proportional to the number of nodal variables. The algorithm uses our in a sense optimal MPRGP algorithm for the solution of strictly convex bound constrained quadratic programming problems with optional preconditioning by the conjugate projector to the subspace defined by the trace of the rigid body motions on the artificial subdomain interfaces. The proof of optimality combines the convergence theory of our MPRGP algorithm, the classical bounds on the spectrum of the mass and stiffness matrices, and our theory of the preconditioning by a conjugate projector for nonlinear problems. The results are confirmed by numerical solution of 2D and 3D dynamic contact problems.

Meshless methods have a number of virtues in problems concerning crack growth and propagation, strain localization, dynamic shear band propagation, projectile penetration, among others. The main drawback of such methods is that the resulting matrices are more densely populated and the computational cost for the formulation and solution of the problem is much higher than the conventional FEM. This is the reason for their limited application to academic and small-scale problems until now. In this paper, we introduce a novel approach for reducing the computational cost of meshless element free Galerkin (EFG) methods by employing domain decomposition techniques on the physical as well as on the algebraic domains. Specifically, we propose the implementation of the dual domain decomposition FETI family of methods that have been successfully used with FEM in many problems in computational mechanics. We address the issues of dividing an overlapping meshfree EFG domain to several overlapping subdomains, following the algebraic decomposition of the FETI method and alternatively on non-overlapping subdomains by modifying the required displacement compatibility conditions and by improving the accuracy of the solution through a stress update procedure.

A general framework of a two-level nonoverlapping Schwarz algorithm for the Stokes problem is developed by relaxing average zero condition on pressure unknowns. This framework allows both discontinuous and continuous pressure finite element spaces. The coarse problem is built by algebraic manipulation after selecting appropriate primal unknowns just like in BDDC algorithms. Performance of the suggested algorithm is presented depending on the selection of finite elements and primal unknowns. Under the same set of primal unknowns, the algorithm for the case with discontinuous pressure functions outperforms one with continuous pressure functions. For the two-dimensional Stokes problem, the algorithm with a set of primal unknowns consisting of velocity unknowns at corners, averages of velocity components over common edges, and pressure unknowns at corners presents good scalability when continuous pressure test functions are used. In both two- and three-dimensional Stokes problems, an improvement can be made for the case with continuous pressure test functions by applying the suggested algorithm to the interface problem, which is obtained by eliminating velocity unknowns and pressure unknowns interior to each subdomains.

International audience; Modeling large systems usually requires metamodels or response surfaces (RSs) of sub-systems. These metamodels are using sampling points in a parameter domain and related responses provided by the solution of parametric partial differential equations (PDEs). Between sampling points the responses are interpolated. We propose to incorporate the RS approximation into the weak form of parametric partial differential equations (PDEs). Hence a multidimensional model-reduction can be achieved. RSs provide very fast predictions. But their enrichment, by adding new sampling points, requires new response evaluations, and therefore new solutions of PDEs. Although it is off-line computations, their complexity does not facilitates the study of large-scale non-linear problems involving a large number of parameters. Reduced-order models can facilitate the solution of the PDEs used to enrich the RSs. We propose a multidimensional a priori model-reduction method to generate or to enrich RSs. It is coined multidimensional because the fields to forecast are defined over an augmented domain in terms of dimension. They are functions of both space variables and parameters that simultaneously evolve in time. This changes the functional space related to the weak form of the PDEs and the definition of the reduced-bases. It has a significant impact on the proposed model-reduction method. In particular, the variable interpolation in the framework of reduced-basis approximations has to be reconsidered. Moreover, a multidimensional reduced integration domain (MRID) is proposed to reduce the complexity of the reduced formulation. It is a subdomain of the full multidimensional domain. The multidimensional hyper-reduction method extracts from the MRID truncated equilibrium equations, truncated residuals and a truncated error indicator. This work is an extension of the a priori hyper-reduction (APHR) method to parametric PDEs coupled to a design of experiments (DOE) method. In this paper, the outputs of the metamodels are the elastic stiffness and the damping coefficient of non-linear mechanical sub-systems that could be incorporated in the model of aircrafts or cars subjected to vibrations. The proposed method has been designed to account for various recent and ongoing research on mathematical formulation of mechanical constitutive equations in material science. Here, we are using a constitutive model proposed by Qi and Boyce for polymers. The three following issues are addressed : Is the multidimensional APHR method more efficient than the APHR method applied individually on separated simulations? What is the efficiency of the proposed approach when adding sampling points in the current parameter domain? What is the efficiency of the method when adding a new dimension to the parameter domain, i.e. when adding a new parameter to enrich a former parametric study?

Based on two-grid discretizations and domain decomposition, a parallel Oseen-linearized finite element algorithm for the stationary Navier–Stokes equations with moderate or large viscosity parameter is proposed and analyzed. The key idea of the algorithm is to first solve a nonlinear problem by Picard iterative method on a coarse grid, and then to solve an Oseen problem in parallel on a fine grid to correct the coarse grid solution. By using local a priori error estimate for the finite element solution and under the uniqueness condition, error bounds of the corresponding finite element solution are analyzed. Numerical results are also given to demonstrate the high efficiency of the algorithm.

In this paper we formulate and analyze two non-conforming high order strategies for the approximation of elastic wave problems in heterogeneous media, namely the Mortar Spectral Element Method and the Discontinuous Galerkin Spectral Element Method. Starting from a common variational formulation we make a full comparison of the two techniques from the points of view of accuracy, convergence, grid dispersion and stability.

The past 10–15 years have seen important extensions of the mortar method, a technique for joining dissimilar grids popularized by the domain decomposition community, to the more general problem of contact and impact interactions in finite element analysis. This development has taken place largely in response to several long-standing problems in computational contact mechanics: lack of robustness in solution of the nonlinear and nonsmooth equations of evolution; degradation of spatial convergence rates in problems involving nonconforming meshes on interfaces; lack of a variationally consistent technique for stress recovery on interfaces; and so on. This survey paper summarizes some of the major steps in development of mortar contact formulations. It begins with a basic summary of the mortaring idea in the context of tied contact, it discusses key concepts required for the extension of these methods to large deformation, large sliding formulations of contact-impact, and it previews new results where lessons learned from mortar contact formulations can be extended to a much broader class of interface mechanics applications, considering in particular enriched interface formulations and embedded interface approaches to fluid–structure interaction.

A Total FETI (TFETI) based domain decomposition algorithm with preconditioning by a natural coarse grid defined by the rigid body motions of subdomains is adapted to the solution of multibody contact problems with friction in 3D and proved to be scalable for Tresca friction. The analysis admits “floating” bodies. The algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition and the discretization parameters is bounded. The analysis is based on the classical results by Farhat et al. on scalability of FETI for linear problems and on our development of optimal quadratic programming algorithms. The algorithm preserves parallel scalability of the classical FETI method. Both theoretical results and numerical experiments indicate that our algorithm is effective. The effectiveness of the method in the inner loop of the fixed point iterations for the solution of the contact problems with Coulomb’s friction is illustrated on the analysis of a mine support.

We investigate multilevel Schwarz domain decomposition preconditioners, to efficiently solve linear systems arising from numerical discretizations of elliptic partial differential equations by the finite element method. In our analysis we deal with unstructured mesh partitions and with subdomain boundaries resulting from using the mesh partitioner. We start from two-level preconditioners with either aggregative or interpolative coarse level components, then we focus on a strategy to increase the number of levels. For all preconditioners, we consider the additive residual update and its multiplicative variants within and between levels. Moreover, we compare the preconditioners behaviour, regarding scalability and rate of convergence. Numerical results are provided for elliptic boundary value problems, including a convection-diffusion problem when suitable stabilization becomes necessary. (C) 2011 Elsevier B.V. All rights reserved.