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A robust adaptive algebraic multigrid linear solver for structural mechanics
Abstract
The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size linear systems, especially when accurate results are sought for derived variables, like stress or deformation fields. Such a task represents the most time-consuming kernel, and motivates the development of robust and efficient linear solvers for these applications. On the one hand, direct solvers are robust and easy to use, but their computational complexity in the best scenario is superlinear, which limits applicability according to the problem size. On the other hand, iterative solvers, in particular those based on algebraic multigrid (AMG)preconditioners, can reach up to linear complexity, but require more knowledge from the user for an efficient setup, and convergence is not always guaranteed, especially in ill-conditioned problems. In this work, we present a novel AMG method specifically tailored for ill-conditioned structural problems. It is characterized by an adaptive factored sparse approximate inverse (aFSAI)method as smoother, an improved least-squared based prolongation (DPLS)and a method for uncovering the near-null space that takes advantage of an already existing approximation. The resulting linear solver has been applied in the solution of challenging linear systems arising from real-world linear elastic structural problems. Numerical experiments prove the efficiency and robustness of the method and show how, in several cases, the proposed algorithm outperforms state-of-the-art AMG linear solvers. Even more important, the results show how the proposed method gives good results even assuming a default setup, making it fully adoptable also for non-expert users and commercial software.
... It is, therefore, natural to have a growing demand towards the development of sophisticated models of increasing size, which are computationally intensive and require better and better performances. A key factor in this sense is the linear solver, which is usually by far the most time-consuming component in a real-world simulation [33,34]. ...
... Remark 3. The computation of |Ω k | and K k in equation (34) can be carried out using the information arising from the discretization grid and the material properties. A more general algebraic strategy, however, can be implemented by following the ideas sketched in [103]. ...
... which provides a fully algebraic interpretation of the classical fixed-stress contribution (34). ...
A preconditioning framework for the coupled problem of frictional contact mechanics and fluid flow in the fracture network is presented. The porous medium is discretized using low-order continuous finite elements, with cell-centered Lagrange multipliers and pressure unknowns used to impose the constraints and solve the fluid flow in the fractures, respectively. This formulation does not require any interpolation between different fields, but is not uniformly inf-sup stable and requires a stabilization. For the resulting 3 x 3 block Jacobian matrix, we design scalable preconditioning strategies, based on the physically-informed block partitioning of the unknowns and state-of-the-art multigrid preconditioners. The key idea is to restrict the system to a single-physics problem, approximately solve it by an inner algebraic multigrid approach, and finally prolong it back to the fully-coupled problem. Two different techniques are presented, analyzed and compared by changing the ordering of the restrictions. Numerical results illustrate the algorithmic scalability, the impact of the relative number of fracture-based unknowns, and the performance on a real-world problem.
... see, for instance, [18] for a short explanation. Generally, the smoother is given by a simple pointwise relaxation method such as (block) Jacobi or Gauss--Seidel, with the second one often preferred even though its use on parallel computers is not straightforward. ...
... However, since the near-kernel of A is related to the smallest eigenpairs of the generalized eigenproblem [9] (9) A\bfitvar = \lambda \bfitvar , a better way to extract an effective test space could be by relying on an iterative eigensolver [22]. In the present implementation, we opt for the simultaneous Rayleigh quotient minimization (SRQCG) [18], whose cost per iteration is only slightly higher than a smoothing step. By contrast, SRQCG can provide a much better approximation of the smallest eigenpairs especially if a good preconditioner is provided. ...
... In turn, large jumps in P introduce high frequencies in the next level operator that the smoother hardly handles. To overcome these difficulties, we compute our BAMG interpolation with an adaptive procedure similar to those described in [18,33]. More specifically, let us define the matrix \Phi whose entries \varphi ij correspond to the jth component of the ith test vector v i for any j in the interpolatory set. ...
... In this work, we propose Chronos a massively parallel implementation of a novel AMG framework [12,6] which is able to adapt all of its components to the problem at hand, from the smoother set-up, to the coarse grid hierarchy and prolongation definition. This is achieved by guessing and iteratively improving in a bootstrap fashion the near-null space of the system, which allows for both testing the smoother and the prolongation operator as well as for inferring the connection strengths between system unknowns. ...
... with I the identity matrix and ω a relaxation factor necessary whenever ρ(M −1 A) > 2, see e.g. [6] for an explanation. Typilly, the smoother is given by a simple pointwise relaxation method such as (block) ...
... By distinction to Gauss-Seidel smoother, aFSAI application is perfectly parallel also in the application as, giving an explicit approximation of the system inverse, it can be applied simply by a matrix-vector product. The price to pay for the use of aFSAI is a not always negligible set-up cost that is usually compensated by a faster covergence, especially in ill-conditioned problems [12,6], where standard smoother fail in dumping high frequencies. ...
... Inspired by the works of Amir et al. (2014) and Peetz and Elbanna (2021), we allow the use of both a geometric and an algebraic multigrid method as a preconditioner for the conjugate gradient method. However, our algebraic multigrid version is different from the one used in Peetz and Elbanna (2021) and is based on the work of Franceschini et al. (2019) and Paludetto . Furthermore, to combine multigrid with the higher order finite element method, our geometric multigrid preconditioner is similar to the h-multigrid cited in Sundar et al. (2015). ...
... The algebraic multigrid (AMG) can be applied to a wider variety of problems, including those with irregular and unstructured meshes, since it is based only on the algebraic equations of the linear systems. In our algebraic multigrid implementation, the coarse mesh is generated using the classic process described in Trottenberg et al. (2001), but with the connections between nodes calculated following a more recent alternative method, proposed by Paludetto and improved by Franceschini et al. (2019), which promises to be more efficient in the case of structural problems. In this alternative approach, the strength of connection between the nodes of the algebraic mesh are obtained based on a test space, constructed using a method based on the minimization of the Rayleigh quotient by conjugate gradients. ...
Designing the topology of three-dimensional structures is a challenging problem due to its memory and time consumption. In this paper, we present a robust and efficient algorithm for solving large-scale 3D topology optimization problems. The robustness of the algorithm is ensured by adopting a globally convergent sequential linear programming method with a stopping criterion based on the first-order optimality conditions of the nonlinear problem. To increase the algorithm’s efficiency, it is combined with a multiresolution scheme that employs different discretizations to deal with displacement, design, and density variables. In addition, the time spent solving the linear equilibrium systems is substantially reduced using multigrid as a preconditioner for the conjugate gradient method. Since multiresolution can lead to the appearance of unwanted artefacts in the structure, we propose an adaptive strategy for increasing the degree of the displacement elements, with a technique for suppressing unnecessary variables that provides accurate solutions with a moderate impact on the algorithm’s performance. We also propose a new thresholding strategy, based on gradient information, to obtain structures composed only by solid or void regions. Computational experiments carried out in Matlab prove that the new algorithm effectively generates high-resolution structures at a low computational cost.
... Many AMG algorithms have been proposed, such as classical AMG, 13,14 smoothed aggregation AMGs, 15,16 AMGs based on element interpolation (AMGe), 17 and element-free AMGe. 18 The adaptive smoothed AMG ( AMG), 19 bootstrap AMG, 20,21 and adaptive smoothing and prolongation-based AMG (aSP-AMG) 22,23 were designed for the ill-conditioned system when the classical AMG algorithms had poor performance and failed to converge. In AMG algorithms, the operator complexity greatly influences the parallel performance. ...
... where R is the restricted operator mapping the vector values from the inner DOFs to the owned DOFs. Equations (23) and (24) indicate that the DOFs should evolve to satisfy the local equations. Although the local balance subspace needs to solve an inverse problem, it can greatly enhance convergence. ...
Solving linear equations and finding eigenvalues are essential tasks in many simulations for engineering applications, but these tasks often cause performance bottlenecks. In this work, the hierarchical subspace evolution method (HiSEM), a hierarchical iteration framework for solving scientific computing problems with solution locality, is proposed. In HiSEM, the original problem is converted to a corresponding minimization function. The problem is decomposed into a series of subsystems. Subspaces and their weights are established for the subsystems and evolve in each iteration. The subspaces are calculated based on local equations and knowledge of physical problems. A small‐scale minimization problem determines the weights of the subspaces. The solution system can be hierarchically established based on the subspaces. As the iterations continue, the degrees of freedom gradually converge to an accurate solution. Two parallel algorithms are derived from HiSEM. One algorithm is designed for symmetric positive definite linear equations, and the other is designed for generalized eigenvalue problems. The linear solver and eigensolver performance is evaluated using a series of benchmarks and a tower model with a complex topology. Algorithms derived from HiSEM can solve a super large‐scale problem with high performance and good scalability.
... where A is a symmetric positive definite (SPD) matrix, may represent one of the most, and often the most [11,16], expensive tasks in any numerical application. AMG is a very popular and effective iterative method for the solution of (1). ...
... Generally, we start from an initial approximation, which is given by rigid body modes in elasticity or the constant vector for Poisson problems, and improve this initial guess through a Simultaneous Rayleigh Quotient Minimization by Conjugate Gradients (SRQCG), as introduced by [17] and used by [11]. The last component in AMG is the prolongation operator P. Since the optimality of the multigrid method relies on the complementarity between smoothing and coarse-grid correction, it is of paramount importance that the prolongation operator accurately represents the low-frequency components of the error. ...
... see for instance [21] for a short explanation. Generally, the smoother is given by a simple pointwise relaxation method such as (block) Jacobi or Gauss-Seidel, with the second one often preferred even though its use on parallel computers is not straightforward. ...
... However, since the near kernel of A is related to the smallest eigenpairs of: (9) Aϕ = λϕ a better way to extract an effective test space could be by relying on an iterative eigensolver. In the present implementation, we opt for the simultaneous Rayleigh quotient minimization (SRQM) [8,21] whose cost per iteration is only slightly higher than a smoothing step. By contrast, SRQM can provide a much better approximation of the smallest eigenpairs especially if a good preconditioner is provided. ...
The numerical simulation of the physical systems has become in recent years a fundamental tool to perform analyses and predictions in several application fields, spanning from industry to the academy. As far as large scale simulations are concerned, one of the most computationally expensive task is the solution of linear systems arising from the discretization of the partial differential equations governing the physical processes.This work presents Chronos, a collection of linear algebra functions specifically designed for the solution of large, sparse linear systems on massively parallel computers (https://www.m3eweb.it/chronos/). Its emphasis is on modern, effective and scalable AMG preconditioners for High Performance Computing (HPC). This work describes the numerical algorithms and the main structures of this software suite, especially from the implementation standpoint. Several numerical results arising from practical mechanics and fluid dynamics applications with hundreds of millions of unknowns are addressed and compared with other state-of-the-art linear solvers, proving Chronos efficiency and robustness.
... Chronos provides a classical AMG implementation with a smoother based on an adaptive variant of the factorized sparse approximate inverse preconditioner [36]. For the elasticity matrix, the prolongation is constructed through a least square fit of the rigid body modes and further improved via energy minimization [25]. The approximations S and X of the Schur complements are as in Subsection 5.1.2. ...
In this paper, we study a class of inexact block triangular preconditioners for double saddle-point symmetric linear systems arising from the mixed finite element and mixed hybrid finite element discretization of Biot's poroelasticity equations. We develop a spectral analysis of the preconditioned matrix, showing that the complex eigenvalues lie in a circle of center (1,0) and radius smaller than 1. In contrast, the real eigenvalues are described in terms of the roots of a third-degree polynomial with real coefficients. The results of numerical experiments are reported to show the quality of the theoretical bounds and illustrate the efficiency of the proposed preconditioners used with GMRES, especially in comparison with similar block diagonal preconditioning strategies along with the MINRES iteration.
... These matrices have been chosen because they are particularly difficult for an iterative solver. A complete description of the test cases can be found in [14,19]. Here, we only report in Tables 5.1 and 5.9 the number of rows (n), the total number of nonzeroes (nnz), the average nonzeroes per row (avg nnz), and an estimate of the condition number (\kappa 2 ), along with the origin of each matrix. ...
... It is usually preferred over classical AMG for solving structural problems, as it is easier to accommodate the multidimensional near-kernel subspace represented by the rigid body modes. Nevertheless, the Dynamic Pattern Least Squares (DPLS) prolongation employed in Chronos allows for an accurate and sparse representation of multidimensional subspaces within a classical AMG framework [77]. Another key component of Chronos is the FSAI smoother [33,78], which is extremely helpful for solving ill-conditioned problems. ...
... Both studies indicate that computational efficiency reduces when the number of cores exceeds a limit value and the computing time becomes dominated by intercommunication between nodes. A different approach to scalability is presented in [31] and [32], where improved iterative solver algorithms are presented and applied to the solution of the different complex FE problems. To fully utilise the computational benefits associated with modern computer architecture and improve computational efficiency, a domain decomposition approach developed previously at Imperial College [33] is utilised in the proposed modelling strategy for masonry bridges and viaducts. ...
... The approach involves an iterative procedure for every set of structural parameters, making it computationally intensive. The model reduction techniques are extended to various substructuring frameworks such as substructure coupling, component mode synthesis (CMS), etc., to obtain reduced-order models of large FE models [5,6,[26][27][28][29][30][31]. Hurty [32] introduced the dynamic substructuring method. ...
A major challenge in structural health monitoring (SHM) is the availability of responses at limited degrees of freedom (DOFs). This requires the determination of the transformation parameter to expand the available DOFs to the full size of a model. The present study proposes an improved iterative model reduction algorithm by eliminating the stiffness terms from the transformation equation. The modified equation is a function of measured modal responses and mass matrices. This enables obtaining the unknown responses without repeated evaluations in case of stiffness reductions typically involved in SHM problems. The transformation parameter is solved using an improved Levenberg-Marquardt (LM) based iterative least-squares optimization technique. Specifically, the LM algorithm is enhanced by introducing an adaptive damping term in the least-squares problem. The proposed approach is further integrated into the substructuring scheme so that it can be readily applied for large finite element models. The algorithm is numerically demonstrated by considering a beam and a ten-storey building model using modal data with Gaussian noise. The effectiveness of the proposed reduced-order model is studied for a gradually decreasing number of modes and available responses for various measurement configurations by comparing with the results of the existing model reduction techniques.
... Since the real-world problems we aim to address are increasingly large and complex, it is natural to experience a growing demand towards the development of larger and larger challenging models, which are computationally intensive and require better and better performances. A key factor in this sense is the linear solver, which is usually by far the most time-consuming component in a real-world simulation [37,38]. ...
A preconditioning framework for the coupled problem of frictional contact mechanics and fluid flow in the fracture network is presented. We focus on a blended finite element/finite volume method, where the porous medium is discretized by low-order continuous finite elements with nodal unknowns, cell-centered Lagrange multipliers are used to prescribe the contact constraints, and the fluid flow in the fractures is described by a classical two-point flux approximation scheme. This formulation is consistent, but is not uniformly inf-sup bounded and requires a stabilization. For the resulting 3×3 block Jacobian matrix, robust and efficient solution methods are not available, so we aim at designing new scalable preconditioning strategies based on the physically-informed block partitioning of the unknowns and state-of-the-art multigrid techniques. The key idea is to restrict the system to a single-physics problem, approximately solve it by an inner algebraic multigrid approach, and finally prolong it back to the fully-coupled problem. Two different techniques are presented, analyzed and compared by changing the ordering of the restrictions. Numerical results illustrate the algorithmic scalability, the impact of the relative number of fracture-based unknowns, and the performance on a benchmark problem. The objective of the analysis is to identify the most promising solution strategy.
... 229,230 . The newly-introduced adaptive smoothening prolongation-based multigrid solver (AMG) designed for ill-conditioned nonlinear problems of structural mechanicswhich is a broader problem-space in a part of which fractal-like patterns can emerge-shows robust performance both for benchmark models and real-world calculations (such as finemicrostructure composite-made mechanical tools), a key feature employed in the approach being adaptive factored sparse approximate inverse [231][232][233] . A more general study of conditioning the AMG for elliptic PDEs is conducted in ref. 234 . ...
The complex interplay between chemistry, microstructure, and behavior of many engineering materials has been investigated predominantly by experimental methods. Parallel to the increase in computer power, advances in computational modeling methods have resulted in a level of sophistication which is comparable to that of experiments. At the continuum level, one class of such models is based on continuum thermodynamics, phase-field methods, and crystal plasticity, facilitating the account of multiple physical mechanisms (multi-physics) and their interaction during microstructure evolution. This paper reviews the status of simulation approaches and software packages in this field and gives an outlook towards promising research directions.
... Working on these components gives rise to a considerable number of possible variants. In this regard, AMG methods have attracted a great interest from the scientific community during the last 20 years and are currently object of intense development, see, for instance, [150][151][152][153][154][155][156][157] for a selection of methods. ...
Linear solvers for reservoir simulation applications are the objective of this review. Specifically, we focus on techniques for Fully Implicit (FI) solution methods, in which the set of governing Partial Differential Equations (PDEs) is properly discretized in time (usually by the Backward Euler scheme), and space, and tackled by assembling and linearizing a single system of equations to solve all the model unknowns simultaneously. Due to the usually large size of these systems arising from real-world models, iterative methods, specifically Krylov subspace solvers, have become conventional choices; nonetheless, their success largely revolves around the quality of the preconditioner that is supplied to accelerate their convergence. These two intertwined elements, i.e., the solver and the preconditioner, are the focus of our analysis, especially the latter, which is still the subject of extensive research. The progressive increase in reservoir model size and complexity, along with the introduction of additional physics to the classical flow problem, display the limits of existing solvers. Intensive usage of computational and memory resources are frequent drawbacks in practice, resulting in unpleasantly slow convergence rates. Developing efficient, robust, and scalable preconditioners, often relying on physics-based assumptions, is the way to avoid potential bottlenecks in the solving phase. In this work, we proceed in reviewing principles and state-of-the-art of such linear solution tools to summarize and discuss the main advances and research directions for reservoir simulation problems. We compare the available preconditioning options, showing the connections existing among the different approaches, and try to develop a general algebraic framework.
... To efficiently exploit currently available computing resources [29], solution algorithms have to be parallel and scalable [30]. This is particularly true for linear solver components, which often consume more than half of the total time spent in any simulation [31,32]. ...
We present a family of preconditioning strategies for the contact problem in fractured and faulted porous media. We combine low-order continuous finite elements to simulate the bulk deformation with piecewise constant Lagrange multipliers to impose the frictional contact constraints. This formulation is not uniformly inf-sup stable and requires stabilization. We improve previous work by Franceschini et al. (2020) by introducing a novel jump stabilization technique that requires only local geometrical and mechanical properties. We then design scalable preconditioning strategies that take advantage of the block structure of the Jacobian matrix using a physics-based partitioning of the unknowns by field type, namely displacement and Lagrange multipliers. The key to the success of the proposed preconditioners is a pseudo-Schur complement obtained by eliminating the Lagrange multiplier degrees of freedom, which can then be efficiently solved using an optimal multigrid method. Numerical results, including complex real-world problems, are presented to illustrate theoretical properties, scalability and robustness of the preconditioner. A comparison with other approaches available in the literature is also provided.
... Depending on the choice for the inner preconditioner of the leading block, we distinguish between MCP+AMG and MCP+FSAI in Table 4. For these approaches, the set of user-specified parameters providing the empirically optimal performance, as suggested in the relevant literature [69,[72][73][74], is used. It can be noticed that RACP is able to solve all the test cases with acceptable computational costs. ...
Frictional contact is one of the most challenging problems in computational mechanics. Typically, it is a tough non-linear problem often requiring several Newton iterations to converge and causing troubles also in the solution to the related linear systems. When contact is modeled with the aid of Lagrange multipliers, the impenetrability condition is enforced exactly, but the associated Jacobian matrix is indefinite and needs a special treatment for a fast numerical solution. In this work, a constraint preconditioner is proposed where the primal Schur complement is computed after augmenting the zero block. The name Reverse is used in contrast to the traditional approach where only the structural block undergoes an augmentation. Besides being able to address problems characterized by singular structural blocks, often arising in contact mechanics, it is shown that the proposed approach is significantly cheaper than traditional constraint preconditioning for this class of problems and it is suitable for an efficient HPC implementation through the Chronos parallel package. Our conclusions are supported by several numerical experiments on mid- and large-size problems from various applications. The source files implementing the proposed algorithm are freely available on GitHub.
... Depending on the choice for the inner preconditioner of the leading block, we distinguish between MCP+AMG and MCP+FSAI in Table 4. For these approaches, the set of user-specified parameters providing the empirically optimal performance, as suggested in the relevant literature [63,62,64,59], is used. It can be noticed that RACP is able to solve all the test cases with acceptable computational costs. ...
Frictional contact is one of the most challenging problems in computational mechanics. Typically, it is a tough nonlinear problem often requiring several Newton iterations to converge and causing troubles also in the solution to the related linear systems. When contact is modeled with the aid of Lagrange multipliers, the impenetrability condition is enforced exactly, but the associated Jacobian matrix is indefinite and needs a special treatment for a fast numerical solution. In this work, a constraint preconditioner is proposed where the primal Schur complement is computed after augmenting the zero block. The name Reverse is used in contrast to the traditional approach where only the structural block undergoes an augmentation. Besides being able to address problems characterized by singular structural blocks, often arising in contact mechanics, it is shown that the proposed approach is significantly cheaper than traditional constraint preconditioning for this class of problems and it is suitable for an efficient HPC implementation through the Chronos parallel package. Our conclusions are supported by several numerical experiments on mid- and large-size problems from various applications. The source files implementing the proposed algorithm are freely available on GitHub.
... However, in challenging real-world problems such as those arising from structural mechanics or fluid flow in highly heterogeneous formations, standard AMG solvers may be slow to converge or even fail, so that more advanced approaches are needed. In particular, the use of powerful smoothers based on approximate inverses can be of great help as shown, for instance, in Paludetto and Franceschini et al. (2019). ...
The solution of linear systems of equations is a central task in a number of scientific and engineering applications. In many cases the solution of linear systems may take most of the simulation time thus representing a major bottleneck in the further development of scientific and technical software. For large scale simulations, nowadays accounting for several millions or even billions of unknowns, it is quite common to resort to preconditioned iterative solvers for exploiting their low memory requirements and, at least potential, parallelism. Approximate inverses have been shown to be robust and effective preconditioners in various contexts. In this work, we show how adaptive Factored Sparse Approximate Inverse (aFSAI), characterized by a very high degree of parallelism, can be successfully implemented on a distributed memory computer equipped with GPU accelerators. Taking advantage of GPUs in adaptive FSAI set-up is not a trivial task, nevertheless we show through an extensive numerical experimentation how the proposed approach outperforms more traditional preconditioners and results in a close-to-ideal behavior in challenging linear algebra problems.
... It is the solver chronos, available at the webpage https://www.m3eweb.it/chronos/, which makes use of an enhanced AMG solver, partially based on a FSAI smoother with dynamical nonzero pattern selection [19,20] In Table 8 we reported the results in solving the FD matrix with = 512 for the PCG method accelerated with either the AMG or the FSAI preconditioners, after some trials to select the optimal parameters. Since the setup time to evaluate the preconditioner is rather high for this approach we reported this in the table as setup while the CPU time for the PCG solution is solver . ...
In this note we exploit polynomial preconditioners for the Conjugate Gradient method to solve large symmetric positive definite linear systems in a parallel environment. We put in connection a specialized Newton method to solve the matrix equation X‐1 = A and the Chebyshev polynomials for preconditioning.We propose a simple modification of one parameter which avoids clustering of extremal eigenvalues in order to speed‐up convergence. We provide results on very large matrices (up to 8.6 billion unknowns in a parallel environment) showing the efficiency of the proposed class of preconditioners.
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... However, in challenging real world problems such as those arising from structural mechanics or fluid flow in highly heterogeneous formations, standard AMG solvers may be slow to converge or even fail, so that more advanced approaches are needed. In particular, the use of powerful smoothers based on approximate inverses can be of great help as shown, for instance, in [31,13]. ...
The solution of linear systems of equations is a central task in a number of scientific and engineering applications. In many cases the solution of linear systems may take most of the simulation time thus representing a major bottleneck in the further development of scientific and technical software. For large scale simulations, nowadays accounting for several millions or even billions of unknowns, it is quite common to resort to preconditioned iterative solvers for exploiting their low memory requirements and, at least potential, parallelism. Approximate inverses have been shown to be robust and effective preconditioners in various contexts. In this work, we show how adaptive FSAI, an approximate inverse characterized by a very high degree of parallelism, can be successfully implemented on a distributed memory computer equipped with GPU accelerators. Taking advantage of GPUs in adaptive FSAI set-up is not a trivial task, nevertheless we show through an extensive numerical experimentation how the proposed approach outperforms more traditional preconditioners and results in a close-to-ideal behaviour in challenging linear algebra problems.
... On the other hand, fast numerical solution techniques involving spatial decomposition in the geometric space, such as domain decomposition (DD) solvers, are relatively newer (as compared to CMS methods) and recently have started to be utilized in industrial applications [5][6][7]. The spatial decomposition is attained through a Schurcomplement-based domain decomposition and partitioning. ...
This paper addresses computational aspects in dynamic sub-structuring of built-up structures with uncertainty. Component
mode synthesis (CMS), which is a model reduction technique, has been integrated within the framework of domain
decomposition (DD), so that reduced models of individual sub-systems can be solved with smaller computational cost compared
to solving the full (unreduced) system by DD. This is particularly relevant for structural dynamics applications where the overall
system physics can be captured by a relatively low number of modes. The theoretical framework of the proposed methodology
has been extended for application in stochastic dynamic systems. To limit the number of eigen-value analyses to be performed
corresponding to the random realizations of input parameters, a locally refined high dimensional model representation model
with stepwise least squares regression is presented. Effectively, a bi-level decomposition is proposed, one in the physical space
and the other in the stochastic space. The geometric decomposition in the physical space by the proposed model reduction-based
DD reduces the computational cost of a single analysis of the system and the functional decomposition in the stochastic space
by the proposed meta-model lowers the number of simulations to be performed on the actual system. The results achieved
by solving a finite-element model of an assembled beam structure and a 3D space frame illustrate good performance of the
proposed methodology, highlighting its potential for complex problems.
We propose a fully distributed parallel adaptive generalization of the Ruge–Stuben algebraic multigrid method. In the adaptive multigrid framework, the coarse space and interpolation operator depend on the eigenvector, corresponding to the minimal eigenvalue. The bootstrap process allows to approximate the eigenvector along the setup process. Integrating the eigenvector into the Ruge–Stuben interpolation operator, we introduce the weak and strong decoupling of connections as well as parameterize the coarse space refinement process for the twice-removed interpolation. The distributed method uses single-layer matrix overlap for the coarse space refinement and the twice-removed interpolation, Luby’s parallel maximal independent set method for both the initial coarse space selection as well as the coarse space refinement and a distributed Gauss–Seidel smoother. The solver is composed using S3M framework. We demonstrate the efficiency of the method on a set of elliptic problems, on systems with rescaled matrices and finally we apply the entire method to the system, originating from the two-phase oil recovery problem.
The ambition of zero emission neighborhood has been set out to operate the building components with zero life cycle greenhouse gas emissions. Hydrocarbon heat pumps as the building components with nearly zero greenhouse gas emissions are faced with the problem of simultaneously reducing refrigerant charges and maintaining competitive capacities. These must be resolved by formulating an effective modeling tool for detailed design and optimization. To develop a resistance–capacitance network model, interdisciplinary knowledge (including graph theory, heat and mass transfer, local moving boundary modeling, and algebraic multigrid) is integrated. The model simultaneously considers heat exchanger circuitries with a high detail level and component behaviors at a holistic system level. Based on experimental validation, the model’s error is ∼ 10%. In an R290 heat pump case study, the coupling mechanism of the system’s configuration, working conditions, and refrigerant distribution are analyzed. Results indicate that the refrigerant charge can be reduced by 11.6% at the expense of a 4.2% reduction in the coefficient of performance, by varying the compressor displacement alone. In response to the variation in the system’s refrigerant, 46.8%, 7.85%, and 44.2% of the total amount of refrigerants are allocated to the condenser, evaporator, and compressor, respectively. The distribution of the refrigerant quality in the heat exchanger is also visualized to indicate the direction of refrigerant reduction.
This article has two main objectives: one is to describe some extensions of an adaptive Algebraic Multigrid (AMG) method of the form previously proposed by the first and third authors, and a second one is to present a new software framework, named BootCMatch, which implements all the components needed to build and apply the described adaptive AMG both as a stand-alone solver and as a preconditioner in a Krylov method. The adaptive AMG presented is meant to handle general symmetric and positive definite (SPD) sparse linear systems, without assuming any a priori information of the problem and its origin; the goal of adaptivity is to achieve a method with a prescribed convergence rate. The presented method exploits a general coarsening process based on aggregation of unknowns, obtained by a maximum weight matching in the adjacency graph of the system matrix. More specifically, a maximum product matching is employed to define an effective smoother subspace (complementary to the coarse space), a process referred to as compatible relaxation, at every level of the recursive two-level hierarchical AMG process.
Results on a large variety of test cases and comparisons with related work demonstrate the reliability and efficiency of the method and of the software.
A model for additive manufacturing by selective laser melting of a powder bed with application to alumina ceramic is presented. Based on Beer–Lambert law, a volume heat source model taking into account the material absorption is derived. The level set method is used to track the shape of deposed bead. An energy solver is coupled with thermodynamic database to calculate the melting-solidification path. Shrinkage during consolidation from powder to liquid and compact medium is modeled by a compressible Newtonian constitutive law. A semi-implicit formulation of surface tension is used, which permits a stable resolution to capture the liquid/gas interface. The influence of different process parameters on temperature distribution, melt pool profiles and bead shapes is discussed. The effects of liquid viscosity and surface tension on melt pool dynamics are investigated. Three dimensional simulations of several passes are also presented to study the influence of the scanning strategy.
This paper introduces an efficient method to automatically generate and mesh a periodic three- dimensional microstructure for matrix-inclusion composites. Such models are of major importance in the field of computational micromechanics for homogenization purposes utilizing unit cell models. The main focus of this contribution is on the creation of cubic representative volume elements (RVEs) featuring a periodic geometry and a periodic mesh topology suitable for the application of periodic boundary condi- tions in the framework of finite element simulations. Our method systematically combines various mesh- ing tools in an extremely efficient and robust algorithm. The RVE generation itself follows a straightfor- ward random sequential absorption approach resulting in a randomized periodic microstructure. Special emphasis is placed on the discretization procedure to maintain a high quality mesh with as few elements as possible, thus, manageable for computer simulations applicable to high volume concentrations, high number of inclusions and complex inclusion geometries. Examples elucidate the ability of the proposed approach to efficiently generate large RVEs with a high number of anisotropic inclusions incorporating extreme aspect ratios but still maintaining a high quality mesh and a low number of elements.
This paper deals with the numerical simulation of friction stir welding (FSW) processes. FSW techniques are used in many industrial applications and particularly in the aeronautic and aerospace industries, where the quality of the joining is of essential importance. The analysis is focused either at global level, considering the full component to be jointed, or locally, studying more in detail the heat affected zone (HAZ). The analysis at global (structural component) level is performed defining the problem in the Lagrangian setting while, at local level, an apropos kinematic framework which makes use of an efficient combination of Lagrangian (pin), Eulerian (metal sheet) and ALE (stirring zone) descriptions for the different computational sub-domains is introduced for the numerical modeling. As a result, the analysis can deal with complex (non-cylindrical) pin-shapes and the extremely large deformation of the material at the HAZ without requiring any remeshing or remapping tools. A fully coupled thermo-mechanical framework is proposed for the computational modeling of the FSW processes proposed both at local and global level. A staggered algorithm based on an isothermal fractional step method is introduced. To account for the isochoric behavior of the material when the temperature range is close to the melting point or due to the predominant deviatoric deformations induced by the visco-plastic response, a mixed finite element technology is introduced. The Variational Multi Scale method is used to circumvent the LBB stability condition allowing the use of linear/linear P1/P1 interpolations for displacement (or velocity, ALE/Eulerian formulation) and pressure fields, respectively. The same stabilization strategy is adopted to tackle the instabilities of the temperature field, inherent characteristic of convective dominated problems (thermal analysis in ALE/Eulerian kinematic framework). At global level, the material behavior is characterized by a thermo–elasto–viscoplastic constitutive model. The analysis at local level is characterized by a rigid thermo–visco-plastic constitutive model. Different thermally coupled (non-Newtonian) fluid-like models as Norton–Hoff, Carreau or Sheppard–Wright, among others are tested. To better understand the material flow pattern in the stirring zone, a (Lagrangian based) particle tracing is carried out while post-processing FSW results. A coupling strategy between the analysis of the process zone nearby the pin-tool (local level analysis) and the simulation carried out for the entire structure to be welded (global level analysis) is implemented to accurately predict the temperature histories and, thereby, the residual stresses in FSW.
Additive manufacturing is a technology rapidly expanding on a number of industrial sectors. It provides design freedom and environmental/ecological advantages. It transforms essentially design files to fully functional products. However, it is still hampered by low productivity, poor quality and uncertainty of final part mechanical properties. The root cause of undesired effects lies in the control aspects of the process. Optimization is difficult due to limited modelling approaches. Physical phenomena associated with additive manufacturing processes are complex, including melting/solidification and vaporization, heat and mass transfer etc. The goal of the current study is to map available additive manufacturing methods based on their process mechanisms, review modelling approaches based on modelling methods and identify research gaps. Later sections of the study review implications for closed-loop control of the process.
This paper presents a parallel preconditioning method for distributed sparse
linear systems, based on an approximate inverse of the original matrix, that
adopts a general framework of distributed sparse matrices and exploits the
domain decomposition method and low-rank corrections. The domain decomposition
approach decouples the matrix and once inverted, a low-rank approximation is
applied by exploiting the Sherman-Morrison-Woodbury formula, which yields two
variants of the preconditioning methods. The low-rank expansion is computed by
the Lanczos procedure with reorthogonalizations. Numerical experiments indicate
that, when combined with Krylov subspace accelerators, this preconditioner can
be efficient and robust for solving symmetric sparse linear systems.
Comparisons with other distributed-memory preconditioning methods are
presented.
This paper describes the use of BIM models derived from point clouds for structural simulation based on Finite Element Analysis
(FEA). Although BIM interoperability has reached a significant level of maturity, the density of laser point clouds provides very
detailed BIM models that cannot directly be used in FEA software. The rationalization of the BIM towards a new finite element
model is not a simple reduction of the number of nodes. The interconnections between the different elements and their materials
require a particular attention: BIM technology includes geometrical aspects and structural considerations that allow one to
understand and replicate the constructive elements and their mutual interaction. The information must be accurately investigated to
obtain a finite element model suitable for a complete and detailed structural analysis. The aim of this paper is to prove that a drastic
reduction of the quality of the BIM model is not necessary. Geometric data encapsulated into dense point clouds can be taken into
consideration also for finite element analysis.
This article proposes a fully three-dimensional finite element model, developed at the meso-scale level, to predict the progressive damage behavior of a single-layer triaxially braided composite subjected to tensile loading conditions. An anisotropic damage model is established by Murakami–Ohno damage theory to predict damage initiation and progression in the fiber tows. A traction–separation law has been applied to predict theoretically the progressive damage of fiber tow interfaces. The proposed model correlates well with experiment on both global stress–strain responses and local strain distributions. According to the damage contours at different global strain levels, the damage development of fiber tows and interlaminar delamination damage of interface are obtained, explicitly analyzed and correlated with experimental observations. The comparison of model prediction and experimental observations indicate that the model can accurately simulate the damage development of this composite material, i.e. fiber bundle splitting, interaction of free-edge effect and delamination, and final failure of the specimen. This paper also discusses the role of material properties/parameters on the global responses through numerical parameter studies.
In this article we introduce new bounds on the effective condition number of deflated and preconditioned-deflated symmetric positive definite linear systems. For the case of a subdomain deflation such as that of Nicolaides [SIAM J. Numer. Anal., 24 (1987), pp. 355--365], these theorems can provide direction in choosing a proper decomposition into subdomains. If grid refinement is performed, keeping the subdomain grid resolution fixed, the condition number is insensitive to the grid size. Subdomain deflation is very easy to implement and has been parallelized on a distributed memory system with only a small amount of additional communication. Numerical experiments for a steady-state convection-diffusion problem are included.
The present paper describes a parallel preconditioned algorithm for the
solution of partial eigenvalue problems for large sparse symmetric
matrices, on parallel computers. Namely, we consider the
Deflation-Accelerated Conjugate Gradient (DACG) algorithm accelerated
by factorized-sparse-approximate-inverse- (FSAI-) type
preconditioners. We present an enhanced parallel implementation of the
FSAI preconditioner and make use of the recently developed Block
FSAI-IC preconditioner, which combines the FSAI and the Block
Jacobi-IC preconditioners. Results onto matrices of large size arising
from finite element discretization of geomechanical models reveal that
DACG accelerated by these type of preconditioners is competitive with
respect to the available public parallel hypre package,
especially in the computation of a few of the leftmost eigenpairs. The
parallel DACG code accelerated by FSAI is written in MPI-Fortran 90
language and exhibits good scalability up to one thousand
processors.
Large discontinuities in material properties, such as encountered in composite materials, lead to ill-conditioned systems of linear equations. These discontinuities give rise to small eigenvalues that may negatively affect the convergence of iterative solution methods such as the Preconditioned Conjugate Gradient (PCG) method. This paper considers the Deflated Preconditioned Conjugate Gradient (DPCG) method for solving such systems. Our deflation technique uses as the deflation space the rigid body modes of sets of elements with homogeneous material properties. We show that in the deflated spectrum the small eigenvalues are mapped to zero and no longer negatively affect the convergence. We justify our approach through mathematical analysis and we show with numerical experiments on both academic and realistic test problems that the convergence of our DPCG method is independent of discontinuities in the material properties.
Elastic–plastic transitions were investigated in three-dimensional (3D) macroscopically homogeneous materials, with microscale randomness in constitutive properties, subjected to monotonically increasing, macroscopically uniform loadings. The materials are cubic-shaped domains (of up to 100 × 100 × 100 grains), each grain being cubic-shaped, homogeneous, isotropic and exhibiting elastic–plastic hardening with a J2 flow rule. The spatial assignment of the grains’ elastic moduli and/or plastic properties is a strict-white-noise random field. Using massively parallel simulations, we find the set of plastic grains to grow in a partially space-filling fractal pattern with the fractal dimension reaching 3, whereby the sharp kink in the stress–strain curve of individual grains is replaced by a smooth transition in the macroscopically effective stress–strain curve. The randomness in material yield limits is found to have a stronger effect than that in elastic moduli. The elastic–plastic transitions in 3D simulations are observed to progress faster than those in 2D models. By analogy to the scaling analysis of phase transitions in condensed matter physics, we recognize the fully plastic state as a critical point and, upon defining three order parameters (the ‘reduced von-Mises stress’, ‘reduced plastic volume fraction’ and ‘reduced fractal dimension’), three scaling functions are introduced to unify the responses of different materials. The critical exponents are universal regardless of the randomness in various constitutive properties and their random noise levels.
We propose a new general algorithm for constructing interpolation weights in al- gebraic multigrid (AMG). It exploits a proper extension mapping outside a neighborhood about a fine degree off reedom (dof ) to be interpolated. The extension mapping provides boundary values (based on the coarse dofs used to perform the interpolation) at the boundary of the neighborhood. The interpolation value is then obtained by matrix dependent harmonic extension ofthe boundary values into the interior ofthe neighborhood. We describe the method, present examples ofuseful extension operators, provide a two-grid anal- ysis ofmodel problems, and, by way ofnumerical experiments, demonstrate the successful application ofthe method to discretized elliptic problems.
Substantial effort has been focused over the last two decades on developing multilevel iterative methods capable of solving the large linear systems encountered in engineering practice. These systems often arise from discretizing partial differential equations over unstructured meshes, and the particular parameters or geometry of the physical problem being discretized may be unavailable to the solver. Algebraic multigrid (AMG) and mul- tilevel domain decomposition methods of algebraic type have been of particular interest in this context because of their promises of optimal performance without the need for ex- plicit knowledge of the problem geometry. These methods construct a hierarchy of coarse problems based on the linear system itself and on certain assumptions about the smooth components of the error. For smoothed aggregation (SA) multigrid methods applied to discretizations of elliptic problems, these assumptions typically consist of knowledge of the near-kernel or near-nullspace of the weak form. This paper introduces an extension of the SA method in which good convergence properties are achieved in situations where explicit knowledge of the near-kernel components is unavailable. This extension is accomplished in an adaptive process that uses the method itself to determine near-kernel components and adjusts the coarsening processes accordingly.
We introduce spectral AMGe (ρAMGe), a new algebraic multigrid method for solving systems of algebraic equations that arise in Ritz-type finite element discretizations of partial differential equations. The method requires access to the element stiffness matrices, which enables accurate approximation of algebraically "smooth" vectors (i.e., error components that relaxation cannot effectively eliminate). Most other algebraic multigrid methods are based in some manner on predefined concepts of smoothness. Coarse-grid selection and prolongation, for example, are often defined assuming that smooth errors vary slowly in the direction of "strong" connections (relatively large coefficients in the operator matrix). One aim of ρAMGe is to broaden the range of problems to which the method can be successfully applied by avoiding any implicit premise about the nature of the smooth error. ρAMGe uses the spectral decomposition of small collections of element stiffness matrices to determine local representations of algebraically smooth error components. This provides a foundation for generating the coarse level and for defining effective interpolation. This paper presents a theoretical foundation for ρAMGe along with numerical experiments demonstrating its robustness. 1. Introduction. Computational investigation is a vital tool for today's scientists and engineers. Modern computer simulation methods demand increasingly greater speed and accuracy, and are being applied to extraordinarily large systems of equations, with tens or hundreds of millions of unknowns. To solve these systems with the accuracy and speed required, massively parallel computers, and algo- rithms that effectively exploit their power, are essential. As problems grow larger, algorithms whose performance scales linearly with the problem size are necessary. Unfortunately, many of today's com- putational simulations use algorithms that do not scale in this sense. Multigrid methods are scalable for many regular-grid problems. However, they can be extremely difficult to devise for the large unstructured grids that many simulations require. Algebraic Multigrid (AMG) overcomes this difficulty by abstracting, in an algebraic sense, the properties that make geo- metric multigrid methods effective. Ideally, this results in a method that is automatic and robust. The classical formulation of AMG grew from the efforts of Brandt, McCormick, Ruge, and Stuben in the 1980's. (For details, see (2, 1, 17).) Interest in AMG methods is growing rapidly, both in academia and in industry, because these methods have great potential for solving the large-scale problems common to many modern applications.
We prove an abstract convergence estimate for the Algebraic Multigrid Method with prolongator defined by a disaggregation
followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes of the same problem
but with natural boundary conditions. The construction is described in the case of a general elliptic system. The condition
number bound increases only as a polynomial of the number of levels, and requires only a uniform weak approximation property
for the aggregation operators. This property can be a-priori verified computationally once the aggregates are known. For illustration,
it is also verified here for a uniformly elliptic diffusion equations discretized by linear conforming quasiuniform finite
elements. Only very weak and natural assumptions on the hierarchy of aggregates are needed.
The numerical simulation of modern engineering problems can easily incorporate millions or even billions degrees of freedom. In several applications, these simulations require the solution to sparse linear systems of equations, and algebraic multigrid (AMG) methods are often standard choices as iterative solvers or preconditioners. This happens due to their high convergence speed guaranteed even in large size problems, which is a consequence of the AMG ability of reducing particular error components across their multilevel hierarchy. Despite carrying the name “algebraic”, most of these methods still rely on additional information other than the global assembled sparse matrix, as for instance the knowledge of the operator near kernel. This fact somewhat limits their applicability as black-box solvers. In this work, we introduce a novel AMG approach featuring the adaptive Factored Sparse Approximate Inverse (aFSAI) method as a flexible smoother as well as three new approaches to adaptively compute the prolongation operator. We assess the performance of the proposed AMG through the solution of a set of model problems along with real-world engineering test cases. Moreover, comparisons are made with the aFSAI and BoomerAMG preconditioners, showing that our new method proves to be superior to the first strategy and comparable to the second one, if not better as in the elasticity problems.
Consequences of submarine landslides include both their direct impact on offshore infrastructure, such as subsea electric cables and gas/oil pipelines, and their indirect impact via the generated tsunami. The simulation of submarine landslides and their consequences has been a long-standing challenge majorly due to the strong coupling among sliding sediments, seawater and infrastructure as well as the induced extreme material deformation during the complete process. In this paper, we propose a unified finite element formulation for solid and fluid dynamics based on a generalised Hellinger-Reissner variational principle so that the coupling of fluid and solid can be achieved naturally in a monolithic fashion. In order to tackle extreme deformation problems, the resulting formulation is implemented within the framework of the particle finite element method. The correctness and robustness of the proposed unified formulation for single-phase problems (e.g. fluid dynamics problems involving Newtonian/Non-Newtonian flows and solid dynamics problems) as well as for multi-phase problems (e.g. two-phase flows) are verified against benchmarks. Comparisons are carried out against numerical and analytical solutions or experimental data that are available in the literature. Last but not least, the possibility of the proposed approach for modelling submarine landslides and their consequences is demonstrated via a numerical experiment of an underwater slope stability problem. It is shown that the failure and post-failure process of the underwater slope can be predicted in a single simulation with its direct threat to a nearby pipeline and indirect threat by generating tsunami being estimated as well.
This chapter outlines the methodology to numerically solve the equations which describe the thermal and mechanical equations for AM processes. This discussion follows that used by the Netfabb Simulation tool which was used to complete the FE validation studies in the rest of the book. However, the modeling approach is general, and may be applied to any AM code. First the necessity of using the non-linear finite element method is shown then a brief explanation of the non-linear FE process is given. The weakly or decoupled modeling method is described, along with reasons for its usefulness and caveats regarding its limitations. The thermal and mechanical models are described with the relevant boundary conditions. A treatment of the primary methods to model the addition of material is given, along with the advantages and limitations of each method. A brief discussion how temperature dependent material properties are applied in the modeling tool is presented. Meshing concerns and methods are then described. Methods of verifying a model is numerically correct by comparing model results to problems with known answers are discussed at length. Finally validation methods are outlined along with error metrics and a discussion of validation criteria.
Fiber reinforced cementitious matrix (FRCM) composites represent a promising alternative to the use of fiber reinforced polymer (FRP) composites for strengthening existing structures. FRCM composites are comprised of a high-strength fiber net (textile) embedded within an inorganic matrix, which is responsible of the stress-transfer between the composite and the substrate. FRCM composites comprising one layer of fiber are generally reported to fail due to debonding of the fibers from the embedding matrix. However, depending on the FRCM material employed, failure at the matrix-substrate interface, interlaminar (delamination) of the matrix, and fiber rupture may also occur. When stitch-bonded textiles are employed, i.e. textiles where longitudinal and transversal bundles are firmly connected, the interlocking between fibers and matrix plays a key role in the matrix-fiber stress-transfer mechanism and failure generally occurs due to cracking of the matrix.
In this paper, the bond behavior of FRCM-masonry joints comprising one layer of a carbon stitch-bonded textile embedded within a lime-based matrix is studied by means of single-lap direct-shear tests of FRCM-masonry joints. A mesoscale three-dimensional finite element approach, which accounts for the matrix-fiber bond behavior and matrix-textile interlocking, is adopted to study the formation and propagation of the matrix cracks in the FRCM strip that eventually led to failure of the FRCM-masonry joints. The finite element analysis was carried out by using a dynamic explicit approach, which allowed for overcoming convergence difficulties associated with severe nonlinearities of the model.
Concrete is non-homogeneous and is composed of three main constituent phases from a mesoscopic viewpoint, namely aggregates, mortar matrix, and interface transition zone (ITZ). A mesoscale model with explicit representation of the three distinctive phases is needed for investigation into the damage processes underlying the macroscopic behaviour of the composite material. This paper presents a full 3-D mesoscale finite element model for concrete. On top of the conventional take-and-place method, an additional process of creating supplementary aggregates is developed to overcome the low packing density problem associated with the take-and-place procedure. An advanced FE meshing solver is employed to mesh the highly unstructured domains. 3D mesoscale numerical simulation is then conducted for concrete specimen under different loading conditions, including dynamic loading with high strain rate. The results demonstrate that detailed mesoscopic damage processes can be realistically captured by the 3D mesoscale model while the macroscopic behaviour compares well with experimental observations under various stress conditions. The well-known inertial confinement effect under dynamic compression can be fully represented with the 3D mesoscale model and the trend of dynamic strength increase with strain rate from the 3D mesoscale analysis agrees well with the experimental data.
This paper presents an original investigation of the seismic cracking behavior of Guandi concrete gravity dam, which is located in the highly seismic zone of China. For this purpose, three dimensional nonlinear finite element analyses are carried out for the Guandi dam-reservoir-foundation system, with the effects of contraction joints and cross-stream seismic excitation considered. The Concrete Damaged Plasticity (CDP) model is utilized to model concrete cracking under seismic loading. The opening/closing and sliding behaviors of contraction joints during earthquake events are modeled using two different surface-to-surface contact models (soft and hard pressure-clearance relationships), which aims to quantify the effect of grouting materials between the joint surfaces. The dynamic interaction between the impounded water and the dam-foundation system is explicitly taken into account by modeling the reservoir water with three dimensional fluid finite elements in the Lagrangian formulation. Several case studies are examined and the results reveal the significant influence of contraction joints and cross-stream ground motions on the dynamic response and damage-cracking risk of the Guandi concrete gravity dam.
In the numerical simulation of structural problems, a crucial aspect concern the solution of the linear system arising from the discretization of the governing equations. In fact, ill-conditioned system, related to an unfavorable eigenspectrum, are quite common in several engineering applications. In these cases the Preconditioned Conjugate Gradient enhanced with the deflation technique seems to be a very promising approach in particular because an effective deflation space is already at hand. In fact, it is possible to utilize rigid body motions of the system, that can be calculated easily and cheaply, and only the knowledge of the geometry of problem is required. This paper investigates the advantages of using a Rigid Body Modes Deflated Conjugate Gradient in the solution of challenging systems arising from structural problems. Two different situations are analyzed: the ill-conditioning caused by low constraining is addressed deflating the total rigid body modes, while the one concerning the heterogeneity of the problem by using the rigid body modes of separate components. Moreover, the implemented method is highly parallel and therefore suitable for High Performance Computing. Numerical results show how both approaches performed successfully in reducing the overall system solution time cost and iterations required for convergence.
This paper is to give an overview of AMG methods for solving large scale systems of equations such as those from the discretization of partial differential equations. AMG is often understood as the acronym of "Algebraic Multi-Grid", but it can also be understood as "Abstract Muti-Grid". Indeed, as it demonstrates in this paper, how and why an algebraic multigrid method can be better understood in a more abstract level. In the literature, there are a variety of different algebraic multigrid methods that have been developed from different perspectives. In this paper, we try to develop a unified framework and theory that can be used to derive and analyze different algebraic multigrid methods in a coherent manner. Given a smoother R for a matrix A, such as Gauss-Seidel or Jacobi, we prove that the optimal coarse space of dimension is the span of the eigen-vectors corresponding to the first eiven-vectors (with ). We also prove that this optimal coarse space can be obtained by a constrained trace-minimization problem for a matrix associated with and demonstrate that coarse spaces of most of existing AMG methods can be viewed some approximate solution of this trace-minimization problem. Furthermore, we provide a general approach to the construction of a quasi-optimal coarse space and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation, and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, the classic AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG, and spectral AMGe.
A class of preconditioners based on balancing domain decomposition by constraints methods is introduced in the Portable, Extensible Toolkit for Scientific Computation (PETSc). The algorithm and the underlying nonoverlapping domain decomposition framework are described with a specific focus on their current implementation in the library. Available user customizations are also presented, together with an experimental interface to the finite element tearing and interconnecting dual-primal methods within PETSc. Large-scale parallel numerical results are provided for the latest version of the code, which is able to tackle symmetric positive definite problems with highly heterogeneous distributions of the coefficients. Current limitations and future extensions of the preconditioner class are also discussed.
In this study, the stress generation caused by phase transitions and lithium intercalation of nickel-manganese-cobalt (NMC) based half cell with realistic 3D microstructures has been studied using finite element method. The electrochemical properties and discharged curves under various C rates are studied. The potential drops significantly with the increase of C rates. During the discharge process, for particles isolated from the conductive channels, several particles with no lithium ion intercalation are observed. For particles in the electrochemical network, the lithium ion concentration increases during the discharge process. The stress generation inside NMC particles is calculated coupled with lithium diffusion and phase transitions. The results show the stresses near the concave and convex regions are the highest. The neck regions of the connected particles can break and form several isolated particles. If the isolated particles are not connected with the electrically conductive materials such as carbon and binder, the capacity loses in battery. For isolated particles in the conductive channel, cracks are more likely to form on the surface. Moreover, stresses inside the particles increase dramatically when considering phase transitions. The phase transitions introduce an abrupt volume change and generate the strain mismatch, causing the stresses increase.
A unified computational method is developed for the modeling of growth in hard and soft biological tissues using a mixture theory approach. The model problem of tissue engineering is considered, whereby a polymeric scaffold is infused by cells and cross-linking proteins. In particular, the underlying scaffold or fibrous network is treated as an inert anisotropic material, and the cross-linking cells or proteins is treated as an isotropic material capable of growth. Both the cases of (i) growth at constant density and (ii) growth at constant volume are considered in order to encompass a broader range of biological response. The relative motions and interactions of the constituents are treated in a generalized sense through the incorporation of mass transfer and drag force terms, in contrast to constrained mixture theory wherein all constituents are constrained to move and deform in unison. Therefore, nodal interpolations are required for both the scaffold and cross-linking solid constituents, thereby modeling concurrent and coexisting constituents. Emphasis herein is placed on the consistent numerical solution procedure for the coupled system of momentum balance equations. Numerical simulations involving growth and relaxation are performed on representative volumes to highlight the features of the method.
In this paper we present a fully distributed, communicator-aware, recursive, and interlevel-overlapped message-passing implementation of the multilevel balancing domain decomposition by constraints (MLBDDC) preconditioner. The implementation highly relies on subcommunicators in order to achieve the desired effect of coarse-grain overlapping of computation and communication, and communication and communication among levels in the hierarchy (namely, interlevel overlapping). Essentially, the main communicator is split into as many nonoverlapping subsets of message-passing interface (MPI) tasks (i.e., MPI subcommunicators) as levels in the hierarchy. Provided that specialized resources (cores and memory) are devoted to each level, a careful rescheduling and mapping of all the computations and communications in the algorithm lets a high degree of overlapping be exploited among levels. All subroutines and associated data structures are expressed recursively, and therefore MLBDDC preconditioners with an arbitrary number of levels can be built while re-using significant and recurrent parts of the codes. This approach leads to excellent weak scalability results as soon as level-1 tasks can fully overlap coarser-levels duties. We provide a model to indicate how to choose the number of levels and coarsening ratios between consecutive levels and determine qualitatively the scalability limits for a given choice. We have carried out a comprehensive weak scalability analysis of the proposed implementation for the three-dimensional Laplacian and linear elasticity problems on structured and unstructured meshes. Excellent weak scalability results have been obtained up to 458,752 IBM BG/Q cores and 1.8 million MPI being, being the first time that exact domain decomposition preconditioners (only based on sparse direct solvers) reach these scales.
The aim of this work is to design and study a Balancing Domain Decomposition by Constraints (BDDC) solver for the nonlinear elasticity system modeling the mechanical deformation of cardiac tissue. The contraction–relaxation process in the myocardium is induced by the generation and spread of the bioelectrical excitation throughout the tissue and it is mathematically described by the coupling of cardiac electro-mechanical models consisting of systems of partial and ordinary differential equations. In this study, the discretization of the electro-mechanical models is performed by Q1 finite elements in space and semi-implicit finite difference schemes in time, leading to the solution of a large-scale linear system for the bioelectrical potentials and a nonlinear system for the mechanical deformation at each time step of the simulation. The parallel mechanical solver proposed in this paper consists in solving the nonlinear system with a Newton–Krylov-BDDC method, based on the parallel solution of local mechanical problems and a coarse problem for the so-called primal unknowns. Three-dimensional parallel numerical tests on different machines show that the proposed parallel solver is scalable in the number of subdomains, quasi-optimal in the ratio of subdomain to mesh sizes, and robust with respect to tissue anisotropy.
Orthodontic tooth movement (OTM) is an adaptive biomechanical response of dentoalveolar components to orthodontic forces, in which remodeling of the alveolar bone occurs in response to changes in the surrounding mechanical environment. In this study, we developed a framework for OTM simulation by combining an image-based voxel finite element method, with a surface-tracking level set method using three-dimensional computer models. For a case study to demonstrate its capability of expressing clinical tooth movement, we observed displacement and rotation of the tooth under three types of force conditions. The simulation results demonstrate that the proposed simulation method has the potential to predict clinical OTM.
The Factorized Sparse Approximate Inverse (FSAI) is an efficient technique for preconditioning parallel solvers of symmetric positive definite sparse linear systems. The key factor controlling FSAI efficiency is the identification of an appropriate nonzero pattern. Currently, several strategies have been proposed for building such a nonzero pattern, using both static and dynamic techniques. This article describes a fresh software package, called FSAIPACK, which we developed for shared memory parallel machines. It collects all available algorithms for computing FSAI preconditioners. FSAIPACK allows for combining different techniques according to any specified strategy, hence enabling the user to thoroughly exploit the potential of each preconditioner, in solving any peculiar problem. FSAIPACK is freely available as a compiled library at http://www.dmsa.unipd.it/~janna/software.html, together with an open-source command language interpreter. By writing a command ASCII file, one can easily perform and test any given strategy for building an FSAI preconditioner. Numerical experiments are discussed in order to highlight the FSAIPACK features and evaluate its computational performance.
This study demonstrates a novel model generation methodology that addresses several limitations of conventional finite element head models (FEHM). By operating chiefly in image space, new structures can be incorporated or merged, and the mesh either decimated or refined both locally and globally. This methodology is employed in the development of a highly bio-fidelic FEHM from high-resolution scan data. The model is adaptable and presented here in a form optimised for impact and blast simulations. The accuracy and feasibility of the model are successfully demonstrated against a widely used experimental benchmark in impact loading and through the investigation of potential brain injury under blast overpressure loading.
Algebraic multigrid (AMG) is a popular and effective solver for systems of linear equations that arise from discretized partial differential equations. While AMG has been effectively implemented on large scale parallel machines, challenges remain, especially when moving to exascale. In particular, stencil sizes (the number of nonzeros in a row) tend to increase further down in the coarse grid hierarchy, and this growth leads to more communication. Thus, as problem size increases and the number of levels in the hierarchy grows, the overall efficiency of the parallel AMG method decreases, sometimes dramatically. This growth in stencil size is due to the standard Galerkin coarse grid operator, P T AP, where P is the prolongation (i.e., interpolation) operator. For example, the coarse grid stencil size for a simple three-dimensional (3D) seven-point finite differencing approximation to diffusion can increase into the thousands on present day machines, causing an associated increase in communication costs. We therefore consider algebraically truncating coarse grid stencils to obtain a non-Galerkin coarse grid. First, the sparsity pattern of the non-Galerkin coarse grid is determined by employing a heuristic minimal “safe” pattern together with strength-of-connection ideas. Second, the nonzero entries are determined by collapsing the stencils in the Galerkin operator using traditional AMG techniques. The result is a reduction in coarse grid stencil size, overall operator complexity, and parallel AMG solve phase times.
The performance of massively parallel direct and iterative methods for solving large sparse systems of linear equations arising in finite element method on unstructured (free) meshes in solid mechanics is evaluated on a latest high performance computing system. We present a comprehensive comparison of a representative group of direct and iterative sparse solvers. Solution time, parallel scalability, and robustness are evaluated on test cases with up to 40 million degrees of freedoms and 3.3 billion nonzeros. The results show that direct solution methods, such as multifrontal with hybrid parallel implementation, as well as new hybrid adaptive block factorized preconditioning iterative methods can take a full advantage of a modern high performance computing system and provide superior solution time and parallel scalability performance.
This paper provides an overview of the main ideas driving the bootstrap
algebraic multigrid methodology, including compatible relaxation and algebraic
distances for defining effective coarsening strategies, the least squares
method for computing accurate prolongation operators and the bootstrap cycles
for computing the test vectors that are used in the least squares process. We
review some recent research in the development, analysis and application of
bootstrap algebraic multigrid and point to open problems in these areas.
Results from our previous research as well as some new results for some model
diffusion problems with highly oscillatory diffusion coefficient are presented
to illustrate the basic components of the BAMG algorithm.
In SIAM J. Numer. Anal. 28 (1991) 1680–1697, Franca and Stenberg developed several Galerkin least squares methods for the solution of the problem of linear elasticity. That work concerned itself only with the error estimates of the method. It did not address the related problem of finding effective methods for the solution of the associated linear systems. In this work, we prove the convergence of a multigrid method. This multigrid is robust in that the convergence is uniform as the parameter ν goes to 1/2. Computational experiments are included.
The choice of the preconditioner is a key factor to accelerate the convergence of eigensolvers for large-size sparse eigenproblems. Although incomplete factorizations with partial fill-in prove generally effective in sequential computations, the efficient preconditioning of parallel eigensolvers is still an open issue. The present paper describes the use of block factorized sparse approximate inverse (BFSAI) preconditioning for the parallel solution of large-size symmetric positive definite eigenproblems with both a simultaneous Rayleigh quotient minimization and the Jacobi–Davidson algorithm. BFSAI coupled with a block diagonal incomplete decomposition proves a robust and efficient parallel preconditioner in a number of test cases arising from the finite element discretization of 3D fluid-dynamical and mechanical engineering applications, outperforming FSAI even by a factor of 8 and exhibiting a satisfactory scalability. Copyright © 2011 John Wiley & Sons, Ltd.
Linear systems arising from discretizations of systems of partial differential equations can be challenging for algebraic multigrid (AMG), as the design of AMG relies on assumptions based on the near-nullspace properties of scalar diffusion problems. For elasticity applications, the near-nullspace of the operator includes the so-called rigid body modes (RBMs), which are not adequately represented by the classical AMG interpolation operators. In this paper we investigate several approaches for improving AMG convergence on linear elasticity problems by explicitly incorporating the near-nullspace modes in the range of the interpolation. In particular, we propose two new methods for extending any initial AMG interpolation operator to exactly fit the RBMs based on the introduction of additional coarse degrees of freedom at each node. Though the methodology is general and can be used to fit any set of near-nullspace vectors, we focus on the RBMs of linear elasticity in this paper. The new methods can be incorporated easily into existing AMG codes, do not require matrix inversions, and do not assume an aggregation approach or a finite element framework. We demonstrate the effectiveness of the new interpolation operators on several 2D and 3D elasticity problems. Copyright © 2009 John Wiley & Sons, Ltd.
A novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented. The spatial domain is partitioned into a set of totally disconnected subdomains, each assigned to an individual processor. Lagrange multipliers are introduced to enforce compatibility at the interface nodes. In the static case, each floating subdomain induces a local singularity that is resolved in two phases. First, the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution. Next, the contributions of these modes are related to the Lagrange multipliers through an orthogonality condition. A parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and Lagrange multipliers, which completes the solution of the problem. When implemented on local memory multiprocessors, this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring. It is also suitable for parallel/vector computers with shared memory. Moreover, unlike parallel direct solvers, it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations.
In this paper we describe an Incomplete LU factorization technique based on a strategy which combines two heuristics. This ILUT factorization extends the usual ILU(O) factorization without using the concept of level of fill-in. There are two traditional ways of developing incomplete factorization preconditioners. The first uses a symbolic factorization approach in which a level of fill is attributed to each fill-in element using only the graph of the matrix. Then each fill-in that is introduced is dropped whenever its level of fill exceeds a certain threshold. The second class of methods consists of techniques derived from modifications of a given direct solver by including a dropoff rule, based on the numerical size of the fill-ins introduced, traditionally referred to as threshold preconditioners. The first type of approach may not be reliable for indefinite problems, since it does not consider numerical values. The second is often far more expensive than the standard ILU(O). The strategy we propose is a compromise between these two extremes.
An algebraic multigrid algorithm for symmetric, positive definite linear systems is developed based on the concept of prolongation by smoothed aggregation. Coarse levels are generated automatically. We present a set of requirements motivated heuristically by a convergence theory. The algorithm then attempts to satisfy the requirements. Input to the method are the coefficient matrix and zero energy modes, which are determined from nodal coordinates and knowledge of the differential equation. Efficiency of the resulting algorithm is demonstrated by computational results on real world problems from solid elasticity, plate bending, and shells.Es wird ein algebraisches Mehrgitterverfahren fr symmetrische, positiv definite Systeme vorgestellt, das auf dem Konzept der gegltteten Aggregation beruht. Die Grobgittergleichungen werden automatisch erzeugt. Wir stellen eine Reihe von Bedingungen auf, die aufgrund der Konvergenztheorie heuristisch motiviert sind. Der Algorithmus versucht diese Bedingungen zu erfllen. Eingabe der Methode sind die Matrix-Koeffizienten und die Starrkrperbewegungen, die aus den Knotenwerten unter Kenntnis der Differentialgleichung bestimmt werden. Die Effizienz des entstehenden Algorithmus wird anhand numerischer Resultate fr praktische Aufgaben aus den Bereichen Elastizitt, Platten und Schalen demonstriert.
The FETI algorithms are a family of numerically scalable domain decomposition methods. They have been designed in the early 1990s for solving iteratively and on parallel machines, large-scale systems of equations arising from the finite element discretization of solid mechanics, structural engineering, structural dynamics, and acoustic scattering problems, and for analyzing complex structures obtained from the assembly of substructures with incompatible discrete interfaces. In this paper, we present the second generation of these methods that operate more efficiently on large numbers of subdomains, offer greater robustness, better performance, and more flexibility for implementation on a wider variety of computational platforms. We also report on the application and performance of these methods for the solution of geometrically non-linear structural analysis problems. We discuss key aspects of their implementation on shared and distributed memory parallel processors, benchmark them against optimized direct sparse solvers, and highlight their potential with the solution of large-scale structural mechanics problems with several million degrees of freedom.
Algebraic multigrid methods are designed for the solution of (sparse) linear systems of equations using multigrid principles. In contrast to standard multigrid methods, AMG does not take advantage of the origin of a particular system of equations at hand, nor does it exploit any underlying geometrical situation. Fully automatically and based solely on algebraic information contained in the given matrix, AMG constructs a sequence of “grids” and corresponding operators. A special AMG algorithm will be presented. For a wide range of problems (including certain problems which do not have a continuous background) this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid methods.
The initial microstructure and local deformation mechanisms of a polyurethane foam during a compression test are investigated by means of X-ray microtomography. A methodology to mesh the actual solid volume is described. The polymer material behaviour is assumed to be elastoplastic. A predictive finite element modelling of the mechanical behaviour of cellular materials is then implemented. The validation of the modelling procedure is performed in relation to the macroscopic mechanical response as well as to the local deformation mechanisms observed during the experiments.