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Algebraic multigrid (AMG) is one of the most widely used solution techniques for linear systems of equations arising from discretized partial differential equations. The popularity of AMG stems from its potential to solve linear systems in almost linear time, that is with an O(n) complexity, where n is the problem size. This capability is crucial at the present, where the increasing availability of massive HPC platforms pushes for the solution of very large problems. The key for a rapidly converging AMG method is a good interplay between the smoother and the coarse-grid correction, which in turn requires the use of an effective prolongation. From a theoretical viewpoint, the prolongation must accurately represent near kernel components and, at the same time, be bounded in the energy norm. For challenging problems, however, ensuring both these requirements is not easy and is exactly the goal of this work. We propose a constrained minimization procedure aimed at reducing prolongation energy while preserving the near kernel components in the span of interpolation. The proposed algorithm is based on previous energy minimization approaches utilizing a preconditioned restricted conjugate gradients method, but has new features and a specific focus on parallel performance and implementation. It is shown that the resulting solver, when used for large real-world problems from various application fields, exhibits excellent convergence rates and scalability and outperforms at least some more traditional AMG approaches. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The Jacobi-Davidson (JD) algorithm is considered one of the most efficient eigensolvers currently available for non-Hermitian problems. It can be viewed as a coupled inner-outer iteration, where the inner one expands the search subspace and the outer one reduces the eigenpair residual. One of the difficulties in the JD efficient use stems from the definition of the most appropriate inner tolerance, so as to avoid useless extra work and keep the number of outer iterations under control. To this aim, the use of an efficient preconditioner for the inner iterative solver is of paramount importance. The present paper describes a fresh implementation of the JD algorithm with controlled inner iterations and block factorized sparse approximate inverse preconditioning for non-Hermitian eigenproblems in a parallel computational environment. The algorithm performance is investigated by comparison with a freely available software package such as SLEPc. The results show that combining the inner tolerance control with an efficient preconditioning technique can allow for a significant improvement of the JD performance, preserving a good scalability. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Adaptive Block FSAI (ABF) is a novel preconditioner which has proved efficient for the parallel solution of symmetric positive definite (SPD) linear systems and eigenproblems. A possible drawback stems from its reduced strong scalability, as the iteration count to converge for a given problem tends to grow with the number of processors used. The preliminary use of graph partitioning techniques can help improve the preconditioner quality and scalability. According to the specific theoretical properties of Block FSAI, different partitionings are selected and tested in a set of matrices arising from SPD engineering applications. The results show that using an appropriate graph partitioning technique with ABF may play an important role to increase the preconditioner efficiency and robustness, allowing for its effective use also in massively parallel simulations. [ABSTRACT FROM AUTHOR]

In recent years the growing popularity of supercomputers has fostered the development of algorithms able to take advantage of the massive parallelism offered by multiple processors. Direct methods, though robust and computationally efficient, hardly exploit high degrees of parallelism. By contrast, Krylov methods preconditioned by Factored Sparse Approximate Inverses (FSAI) provide, at least in principle, a perfectly parallel approach but are often thwarted by an excessive set-up cost. In this paper we extend the concept of supernode from sparse LU factorizations to approximate inverses, and use it to accelerate the computation of an FSAI-type preconditioner. The numerical experiments on real-world problems show that the overall FSAI efficiency can be significantly increased while preserving its intrinsic parallelism. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: The efficient solution to nonsymmetric linear systems is still an open issue, especially on parallel computers. In this paper we generalize to the unsymmetric case the Block Factorized Sparse Approximate Inverse (Block FSAI) preconditioner which has already proved very effective on symmetric positive definite (SPD) problems. Block FSAI is a hybrid approach combining an “inner” preconditioner, with the aim of transforming the system matrix structure to block diagonal, with an “outer” one, a block diagonal incomplete or exact factorization intended to improve the conditioning of each block. The proposed algorithm is experimented with in a number of large size matrices showing both a good robustness and scalability. [Copyright &y& Elsevier]

Adaptive block factorized sparse approximate inverse (FSAI) (ABF) is a novel algebraic preconditioner for the cost-effective parallel solution of symmetric positive definite linear systems. However, a main drawback stems from its reduced scalability, as the iteration count to convergence tends to grow when the number of processors increases. A domain decomposition Schur complement approach can enhance both the ABF performance and scalability. It is demonstrated that the enhanced ABF preconditioner is superior to the native block FSAI, reducing at the same time the construction and communication computational burden. Numerical results from different large size applications show that the proposed algorithm can improve significantly the preconditioner, allowing for its efficient use in massively parallel simulations as well. [ABSTRACT FROM AUTHOR]

SUMMARY 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. [ABSTRACT FROM AUTHOR]

Constraint preconditioners have proved very efficient for the solution of ill-conditioned finite element (FE) coupled consolidation problems in a sequential computing environment. Their implementation on parallel computers, however, is not straightforward because of their inherent sequentiality. The present paper describes a novel parallel inexact constraint preconditioner (ParICP) for the efficient solution of linear algebraic systems arising from the FE discretization of the coupled poro-elasticity equations. The ParICP implementation is based on the use of the block factorized sparse approximate inverse incomplete Cholesky preconditioner, which is a very recent and effective development for the parallel preconditioning of symmetric positive definite matrices. The ParICP performance is experimented with in real 3D coupled consolidation problems, proving a scalable and efficient implementation of the constraint preconditioning for high-performance computing. ParICP appears to be a very robust algorithm for solving ill-conditioned large-size coupled models in a parallel computing environment. [ABSTRACT FROM AUTHOR]

Abstract: Hydrocarbon production, subsurface sequestration, and gas storage in exhausted reservoirs are a few anthropogenic activities involving the extraction or injection of fluids underground. As a consequence, the withdrawn/injected formation undergoes an effective stress change that may induce deep deformations propagating to the ground level with possible fault re-activations. The geomechanical parameters needed to simulate and predict the above processes can also be indirectly inferred by Interferometric Synthetic Apertur Radar (InSAR)-based techniques that are used to monitor the land surface displacements. In the present study, the InSAR measurements above two exhausted fields of the Po River plain, Italy, presently employed for seasonal gas storage, are used to develop, set up, and calibrate a transversely isotropic geomechanical model of the porous medium. It is shown that only assuming transverse isotropy allows for a satisfactory reproduction of both the vertical and horizontal displacements that cannot be simultaneously matched by a simpler isotropic model. Moreover, similar parameters prove correct for both the reservoirs, located about 30km apart, providing evidence of the regional range of the above model applicability. [Copyright &y& Elsevier]

Abstract. An adaptive algorithm is presented to generate automatically the nonzero pattern of the block factored sparse approximate inverse (BFSAI) preconditioner. It is demonstrated that in symmetric positive definite (SPD) problems BFSAI minimizes an upper bound to the Kaporin number of the preconditioned matrix. The mathematical structure of this bound suggests an efficient and easily parallelizable strategy for improving the given nonzero pattern of BFSAI, thus providing a novel adaptive BFSAI (ABF) preconditioner. Numerical experiments performed on large sized finite element problems show that ABF coupled with a block incomplete Cholesky (IC) outperforms BFSAI- IC even by a factor of 4, preserving the same preconditioner density and exhibiting an excellent parallelization degree. [ABSTRACT FROM AUTHOR]

Abstract: The Finite Element Method (FEM) is widely used in civil and mechanical engineering to simulate the behavior of complex structures and, more specifically, to predict stress and deformation fields of structural parts or mechanical bodies. In the former case, the coupling between different types of elements, such as beams, trusses, and shells, is often required, while in the latter fully 3D discretizations are typically used. For both, FEM leads to symmetric positive definite (SPD) matrices that, depending on the type of discretization and especially on the topology of the nodal connections, may be efficiently solved by either the Preconditioned Conjugate Gradient (PCG) or a direct solver such as the routine MA57 of the Harwell Software Library. Numerical experiments are shown and discussed where the effect of spatial discretization, different solution techniques, and a possible nodal reordering, is explored. The PCG preconditioner used is a variant of the incomplete Cholesky factorization with variable fill-in. It is shown that for structures with 1D or 2D connections, such as for example a bridge, MA57 performs usually better than PCG. In this case it is noted that some reorderings specifically designed and implemented for direct elimination methods can be very helpful for PCG as well as they yield a cheaper preconditioner and lead to a much faster PCG convergence. The main disadvantage is the need for an appropriate degree of fill-in for the preconditioner which turns out to be problem dependent and must be found empirically. However, in fully 3D problems, arising for example from the FE discretization of structural components or geomechanical structures, PCG outperforms MA57 while also requiring much less memory, and thus allowing for the use of much refined grids, if needed. With the aid of a large geomechanical problem it is shown that direct solvers may not be (even) used on serial computers due to their prohibitive computational cost with PCG the only viable alternative solver. [Copyright &y& Elsevier]

Abstract: Contact mechanics can be addressed numerically by finite elements using either a penalty formulation or the Lagrange multipliers. The penalty approach leads to a linearized symmetric positive definite system which can prove severely ill-conditioned, with the iterative solution to large 3D problems requiring expensive preconditioners to accelerate, or even to allow for, convergence. In the present paper a mixed constraint preconditioner (MCP) is developed and implemented for the iterative solution of contact problems based on the penalty approach. A theoretical bound for the eigenspectrum of the preconditioned matrix is provided depending on the user-specified parameters that control the preconditioner fill-in. The performance of MCP combined with the preconditioned conjugate gradient is studied in two real 3D test cases describing the geomechanics of faulted porous media and compared to a well-established preconditioner such as the incomplete Cholesky decomposition ILLT with optimal fill-in. Numerical results show that typically MCP outperforms ILLT especially in the most challenging problems where very large penalty parameters must be used. Finally, a modification aimed at simplifying the MCP implementation and increasing its robustness is developed and experimented with, showing that the modified MCP can be viewed as an efficient and viable alternative to both MCP and ILLT, especially in problems where the size of the contact surfaces is large with respect to the global domain. [Copyright &y& Elsevier]

The numerical simulation of modern engineering problems can easily incorporate millions or even billions of unknowns. In several applications, sparse linear systems with symmetric positive definite matrices need to be solved, and algebraic multigrid (AMG) methods represent common choices for the role of iterative solvers or preconditioners. The reason for their popularity relies on the fast convergence that these methods provide even in the solution of large size problems, which is a consequence of the AMG ability to reduce particular error components across their multilevel hierarchy. Despite carrying the name "algebraic," most of these methods still make assumptions on additional information other than the global assembled matrix, such as the knowledge of the operator's near kernel, which 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, we perform comparisons to other methods such as 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 solution of linear elasticity models. [ABSTRACT FROM AUTHOR]

The simulation of the mechanics of geological faults and fractures is of paramount importance in several applications, such as ensuring the safety of the underground storage of wastes and hydrocarbons or predicting the possible seismicity triggered by the production and injection of subsurface fluids. However, the stable numerical modeling of ground ruptures is still an open issue. The present work introduces a novel formulation based on the use of the Lagrange multipliers to prescribe the constraints on the contact surfaces. The variational formulation is modified in order to take into account the frictional work along the activated fault portion according to the principle of maximum plastic dissipation. The numerical model, developed in the framework of the Finite Element method, provides stable solutions with a fast convergence of the non-linear problem. The stabilizing properties of the proposed model are emphasized with the aid of a realistic numerical example dealing with the generation of ground fractures due to groundwater withdrawal in arid regions. [ABSTRACT FROM AUTHOR]

Abstract: Anthropogenic CO2 sequestration in deep geological formations may represent a viable option to fulfil the requirements of the 1997 Kyoto protocol on the reduction of greenhouse gas emissions. Scenarios of CO2 sequestration through three injection wells in an exploited gas field located in the Po sedimentary basin (Italy) are simulated with the final target to understand the geomechanical consequences of the injection of carbon dioxide. Investigated scenarios include, as a hypothetical case, the long-term injection of CO2 until the initial reservoir pressure is exceeded by as much as 40% over a period of about 100years. The process is analyzed from the geomechanical point of view using a finite element–interface element (FE–IE) model with the following main issues addressed: (1) prediction of the possible land vertical uplift and corresponding impact on the ground infrastructures; (2) evaluation of the stress state induced in the reservoir formation with the possible generation of fractures and (3) a risk analysis for the activation of existing faults. The geomechanical constitutive law of the Northern Adriatic basin relying on the radioactive marker interpretation is implemented into the FE model, while an elasto-plastic relationship based on the Mohr–Coulomb criterion is used for the IE reproducing the fault behaviour. The in situ stress prior to the gas field exploitation is compressive with the principal horizontal stress in the direction perpendicular to the major faults equal to the vertical stress. The results show that the ground surface rebound due to the overpressure generated by the CO2 sequestration partially mitigates the land subsidence experienced by the area because of the previous gas field depletion with differential displacements that are confined within the safety bounds suggested in the literature for the surface infrastructures. Activation of a few faults lying close to the northern reservoir boundary points to a slip of a couple of centimetres only and occurs before the CO2 plume reaches the activated faults, so there is little chance for a CO2 escape. The caprock is also proven to maintain its full integrity during the injection process. Finally the shear stress appears to approach the limiting state prone to the rock shear failure exclusively within the reservoir on a quite local scale and can hardly jeopardize the overall effectiveness of the CO2 sequestration. [Copyright &y& Elsevier]

Anthropogenic land subsidence can be evaluated and predicted by numerical models, which are often built over deterministic analyses. However, uncertainties and approximations are present, as in any other modeling activity of real-world phenomena. This study aims at combining data assimilation techniques with a physically-based numerical model of anthropogenic land subsidence in a novel and comprehensive workflow, to overcome the main limitations concerning the way traditional deterministic analyses use the available measurements. The proposed methodology allows to reduce uncertainties affecting the model, identify the most appropriate rock constitutive behavior and characterize the most significant governing geomechanical parameters. The proposed methodological approach has been applied in a synthetic test case representative of the Upper Adriatic basin, Italy. The integration of data assimilation techniques into geomechanical modeling appears to be a useful and effective tool for a more reliable study of anthropogenic land subsidence. [ABSTRACT FROM AUTHOR]

The numerical simulation of 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 tasks is the solution of linear systems of equations arising from the discretization of the partial differential equations governing 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. Its emphasis is on modern, effective, and scalable Algebraic Multigrid (AMG) preconditioners for high performance computing (HPC). This work describes the numerical algorithms and the main structures of this software suite, especially from an 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's efficiency and robustness. [ABSTRACT FROM AUTHOR]

This work discusses a general approach for preconditioning the block Jacobian matrix arising from the discretization and linearization of coupled multiphysics problem. The objective is to provide a fully algebraic framework that can be employed as a starting point for the development of specialized algorithms exploiting unique features of the specific problem at hand. The basic idea relies on approximately computing an operator able to decouple the different processes, which can then be solved independently one from the other. In this work, the decoupling operator is computed by extending the theory of block sparse approximate inverses. The proposed approach is implemented for two multiphysics applications, namely the simulation of a coupled poromechanical system and the mechanics of fractured media. The numerical results obtained in experiments taken from real-world examples are used to analyze and discuss the properties of the preconditioner. • A general framework is introduced for the block preconditioning of discrete systems arising from coupled multi-physics applications. • A decoupling operator is introduced by using the theory of block sparse approximate inverse preconditioners. • The framework is specialized for two applications: (i) coupled poromechanics, (ii) mechanics of fractured media. • Specific techniques for preserving the definiteness of the Schur complements are introduced for each application. • The preconditioners arising from the proposed general framework are used to solve real-world demanding applications. [ABSTRACT FROM AUTHOR]

Summary: A 3D elasto‐plastic rate‐dependent model for rock mechanics is formulated and implemented into a Finite Element (FE) numerical code. The model is based on the approach proposed by Vermeer and Neher (A soft soil model that accounts for creep. In: Proceedings of the International Symposium "Beyond 2000 in Computational Geotechnics," pages 249‐261, 1999). An original strain‐driven algorithm with an Inexact Newton iterative scheme is used to compute the state variables for a given strain increment.The model is validated against laboratory measurements, checked on a simplified test case, and used to simulate land subsidence due to groundwater and hydrocarbon production. The numerical results prove computationally effective and robust, thus allowing for the use of the model on real complex geological settings. [ABSTRACT FROM AUTHOR]

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. • An Adaptive AMG linear solver for ill-conditioned structural problems is presented. • Results on more than 10 realistic examples prove its robustness and high efficiency. • Its performance is comparable or superior to other renowned state-of-art solvers. • It does not require tuning of user-parameters to achieve excellent performances. [ABSTRACT FROM AUTHOR]

The reactivation of faults and the generation of fractures can be caused by stress changes due to injection and/or production of fluids into and/or from the subsurface. The simulation of these processes, which could be associated with (micro-)seismicity, is affected by a high uncertainty. The aim of this work is at developing a mathematical framework to quantify and possibly reduce the prior modeling uncertainties by assimilation of seismic data. The mechanics of fault (re-)activation is simulated by a Finite Element (FE) numerical model where the discontinuous displacements between the fault surfaces are suitably considered using appropriate Interface Elements (IEs). The study is carried out by using a stochastic approach, with a global sensitivity analysis (gSA) based on Sobol' indices initially performed to estimate the influence of the input parameters on the model solution. Then, a Markov Chain Monte Carlo (MCMC) sampling technique based on the generalized Polynomial Chaos expansion (gPC) surrogate solution is used to update the prior information conditioned on seismic observations. The methodology is tested on a 3D synthetic test case. The uncertain input is the natural stress regime and the Mohr-Coulomb parameters characterizing the fault activation criterion. A good reduction of the prior uncertainty is obtained, showing that the assimilation of seismic data can have a promising potential for improving the subsurface characterization. [ABSTRACT FROM AUTHOR]

One of the most time-consuming tasks in the procedures for the numerical study of PDEs is the solution to linear systems of equations. To that purpose, iterative solvers are viewed as a promising alternative to direct methods on high-performance computers since, in theory, they are almost perfectly parallelizable. Their main drawback is the need of finding a suitable preconditioner to accelerate convergence. The factorized sparse approximate inverse (FSAI), mainly in its adaptive form, has proven to be an effective parallel preconditioner for several problems. In the present work, we report about two novel ideas to dynamically compute, on graphics processing units (GPUs), the FSAI sparsity pattern, which is the main task in its setup. The first approach, borrowed from the CPU implementation, uses a global array as a nonzero indicator, whereas the second one relies on a merge-sort procedure of multiple arrays. We will show that the second approach requires significantly less memory and overcomes issues related to the limited global memory available on GPUs. Numerical tests prove that the GPU implementation of FSAI allows for an average speed-up of 7.5 over a parallel CPU implementation. Moreover, we will show that the preconditioner computation is still feasible using single precision arithmetic with a further 20% reduction of the setup cost. Finally, the strong scalability of the overall approach in shown in a multi-GPU setting. [ABSTRACT FROM AUTHOR]

Summary: Factorized sparse approximate inverse (FSAI) preconditioners are robust algorithms for symmetric positive matrices, which are particularly attractive in a parallel computational environment because of their inherent and almost perfect scalability. Their parallel degree is even redundant with respect to the actual capabilities of the current computational architectures. In this work, we present two new approaches for FSAI preconditioners with the aim of improving the algorithm effectiveness by adding some sequentiality to the native formulation. The first one, denoted as block tridiagonal FSAI, is based on a block tridiagonal factorization strategy, whereas the second one, domain decomposition FSAI, is built by reordering the matrix graph according to a multilevel k‐way partitioning method followed by a bandwidth minimization algorithm. We test these preconditioners by solving a set of symmetric positive definite problems arising from different engineering applications. The results are evaluated in terms of performance, scalability, and robustness, showing that both strategies lead to faster convergent schemes regarding the number of iterations and total computational time in comparison with the native FSAI with no significant loss in the algorithmic parallel degree. [ABSTRACT FROM AUTHOR]

The automatic generation of meshes for the Finite Element method can be an expensive computational burden, especially in structural problems with localized stress peaks. The use of meshless methods can address such an issue, as these techniques do not require the existence of an underlying connection among the nodes selected in a general domain. However, a thoroughly meshfree technique can be computationally quite expensive. Usually, the most expensive tasks rely on identifying the nodal contacts and computing the Galerkin integrals. In this thesis we advance a novel hybrid technique that blends Finite Elements with the Meshless Local Petrov-Galerkin method with the aim at exploiting the most attractive properties of each procedure. The idea relies on the use of the Finite Element Method to compute a background solution that is locally improved by enriching the approximating space with the basis functions associated to a few meshless nodes, thus taking advantage of the flexibility ensured by the use of particles disconnected from an underlying grid. Adding the meshless particles only where needed avoids the cost of mesh refining, or even of re-meshing, without the prohibitive burden of a thoroughly meshfree approach. In particular, two enriching methods are introduced and discussed, with applications in structural mechanics. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

The coupled equations of Biot's poroelasticity, consisting of stress equilibrium and fluid mass balance in deforming porous media, are numerically solved. The governing partial differential equations are discretized by an Element-based Finite Volume Method (EbFVM), which can be used in three dimensional unstructured grids composed of elements of different types. One of the difficulties for solving these equations is the numerical pressure instability that can arise when undrained conditions take place. In this paper, a stabilization technique is developed to overcome this problem by employing an interpolation function for displacements that considers also the pressure gradient effect. The interpolation function is obtained by the so-called Physical Influence Scheme (PIS), typically employed for solving incompressible fluid flows governed by the Navier–Stokes equations. Classical problems with analytical solutions, as well as three-dimensional realistic cases are addressed. The results reveal that the proposed stabilization technique is able to eliminate the spurious pressure instabilities arising under undrained conditions at a low computational cost. [ABSTRACT FROM AUTHOR]

The use of factorized sparse approximate inverse (FSAI) preconditioners in a standard multilevel framework for symmetric positive deFInite (SPD) matrices may pose a number of issues as to the deFIniteness of the Schur complement at each level. The present work introduces a robust multilevel approach for SPD problems based on FSAI preconditioning, which eliminates the chance of algorithmic breakdowns independently of the preconditioner sparsity. The multilevel FSAI algorithm is further enhanced by introducing descending and ascending low-rank corrections, thus giving rise to the multilevel FSAI with low-rank corrections (MFLR) preconditioner. The proposed algorithm is investigated in a number of test problems. The numerical results show that the MFLR preconditioner is a robust approach that can significantly accelerate the solver convergence rate preserving a good degree of parallelism. The possibly large set-up cost, mainly due to the computation of the eigenpairs needed by low-rank corrections, makes its use attractive in applications where the preconditioner can be recycled along a number of linear solves. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Graphics Processing Units (GPUs) exhibit significantly higher peak performance than conventional CPUs. However, in general only highly parallel algorithms can exploit their potential. In this scenario, the iterative solution to sparse linear systems of equations could be carried out quite efficiently on a GPU as it requires only matrix-by-vector products, dot products, and vector updates. However, to be really effective, any iterative solver needs to be properly preconditioned and this represents a major bottleneck for a successful GPU implementation. Due to its inherent parallelism, the factored sparse approximate inverse (FSAI) preconditioner represents an optimal candidate for the conjugate gradient-like solution of sparse linear systems. However, its GPU implementation requires a nontrivial recasting of multiple computational steps. We present our GPU version of the FSAI preconditioner along with a set of results that show how a noticeable speedup with respect to a highly tuned CPU counterpart is obtained. [ABSTRACT FROM AUTHOR]

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. • A preconditioner for saddle point matrices arising in contact mechanics is presented. • Theoretical analysis of a novel scalable preconditioner for block matrices are shown. • Numerical results are presented to illustrate strong/weak scalability and robustness. [ABSTRACT FROM AUTHOR]

A Correction to this paper has been published: https://doi.org/10.1007/s10596-021-10079-6 [ABSTRACT FROM AUTHOR]

The reliability of in situ compaction measurements made in producing gas/oil reservoirs with the aid of the radioactive marker technique (RMT) is a very important issue as an essential parameter controlling the geomechanical response of fluid extraction from the subsurface, i.e., the rock vertical compressibility cM, is obtained from RMT data. The validity of cM as provided by RMT has recently been questioned on the basis of the possible disturbance that a fully cased hole can induce on the actual reservoir compaction. By the use of a numerical approach and real casing information, the present paper addresses and discusses the reliability of RMT measurements performed in hole casings of the Northern Adriatic gas fields, Italy. The results show that the casing stiffness affects the RMT measurements only marginally, and especially so at a larger depth (i.e., 3,000 m), and does not significantly impact on the estimate of cM derived from RMT data. [ABSTRACT FROM AUTHOR]