A Parallel Adaptive Block FSAI Preconditioner for Finite Element Geomechanical Models

A numerical study of the efficiency of the modified conjugate gradients (MCG) is performed using different preconditioning schemes. The MCG behavior is evaluated in connection with the solution of large linear sets of symmetric positive definite (p.d.) equations, arising from the finite element (f.e.) integration of partial differential equations of parabolic and elliptic type and the analysis of the leftmost eingenspectrum of the corresponding matrices. A simple incomplete Cholesky factorization ICCG(O) having the same sparsity pattern as the original problem is compared with a more complex technique ICAJ (Ψ) where the triangular factor is allowed to progressively fill in depending on a rejection parameter Ψ. The performance of the preconditioning algorithms is explored on finite element equations whose size N ranges between 150 and 2300. The results show that an optimal Ψopt may be found which minimizes the overall CPU time for the solution of both the linear system and the eigenproblem. The comparison indicates that ICAJ (Ψopt) is not significantly more efficient than ICCG(O), which therefore appears to be a simple, robust, and reliable method for the preconditioning of large sparse finite element models.

In recent years, autonomously guided parachute systems have been developed with the goal of increasing the landing accuracy of airdrop operations. These precision airdrop systems utilize an onboard global positioning system (GPS) coupled with a guidance, navigation and control (GNC) system to continuously monitor and correct the flight path of the parachute system during descent. A critical component of the GNC system is an algorithm that predicts the response of the system to various aerodynamic forces in real-time that are typically based on a low-order multi-degree of freedom (DOF) dynamics model. Reduced order modeling is the process of representing a complex dynamic system with many DOF with a model having relatively few DOF. The goal of this study is to examine the use of large scale finite element models to construct and calibrate Reduced Order Models (ROM) for parachute systems. The resulting ROM would provide parachute designers with a simple tool to examine parachute behavior under various operating conditions. The use of numerical simulations to calibrate the ROM would help to eliminate the number of physical tests needed to characterize a parachute system’s properties. In this study, a 6 degree of freedom ROM is adopted. The system dynamic properties for the ROM and the resultant forces and moments acting on the ROM are obtained from finite element simulations. The 6 DOF model originally proposed by Cockrell and Doherr and later used by Dobrokhodov is adopted for the ROM. The finite element (FE) code, TENSION, is used to perform the large scale simulations and extract the ROM dynamic properties and resultant forces. Several examples are performed that verify the ROM procedure by comparing ROM predictions to FE simulations for various parachute system trajectories. The performance of the ROM is shown to be excellent for in-plane motion under an oscillating wind field. It is shown, however, that the ROM procedure deteriorates as the FE model is refined by adding more DOF. It is demonstrated that, for highly deformable FE models with many DOF, more sophisticated ROM procedures are needed.

Application demands and grand challenges in numerical simulation require for both highly capable computing platforms and efficient numerical solution schemes. Power constraints and further miniaturization of modern and future hardware give way for multi- and manycore processors with increasing fine-grained parallelism and deeply nested hierarchical memory systems -- as already exemplified by recent graphics processing units. Accordingly, numerical schemes need to be adapted and re-engineered in order to deliver scalable solutions across diverse processor configurations. Portability of parallel software solutions across emerging hardware platforms is another challenge. This work investigates multi-coloring and re-ordering schemes for block Gauß-Seidel methods and, in particular, for incomplete LU factorizations with and without fill-ins. We consider two matrix re-ordering schemes that deliver flexible and efficient parallel preconditioners. The general idea is to generate block decompositions of the system matrix such that the diagonal blocks are diagonal itself. In such a way, parallelism can be exploited on the block-level in a scalable manner. Our goal is to provide widely applicable, out-of-the-box preconditioners that can be used in the context of finite element solvers. We propose a new method for anticipating the fill-in pattern of ILU($p$) schemes which we call the power($q$)-pattern method. This method is based on an incomplete factorization of the system matrix $A$ subject to a predetermined pattern given by the matrix power $|A|^(p+1)$ and its associated multi-coloring permutation $. We prove that the obtained sparsity pattern is a superset of our modified ILU($p$) factorization applied to pi A pi^(-1). As a result, this modified ILU($p$) applied to multi-colored system matrix has no fill-ins in its diagonal blocks. This leads to an inherently parallel execution of triangular ILU($p$) sweeps. In addition, we describe the integration of the preconditioners into the HiFlow$^3$ open-source finite element package that provides a portable software solution across diverse hardware platforms. On this basis, we conduct performance analysis across a variety of test problems on multi-core CPUs and GPUs that proves efficiency, scalability and flexibility of our approach. Our preconditioners achieve a solver acceleration by a factor of up to 1.5, 8 and 85 for three different test problems. The GPU versions of the preconditioned solver are by a factor of up to 4 faster than an OpenMP parallel version on eight cores.

PurposeThis paper aims to deal with the verification of local quantities of interest obtained through linear elastic finite element analysis. A technique is presented for determining the most accurate error estimation. This technique enables one to address industrial‐size problems while keeping computing costs reasonable.Design/methodology/approachThe concept of error in constitutive relation is used to assess the quality of the finite element solution. The key issue is the construction of admissible fields. The objective is to show that it is possible to build admissible fields using a new method. These fields are obtained by using a high‐quality construction over a limited zone while the construction is less refined and less expensive elsewhere.FindingsNumerical tests are presented in order to illustrate a very satisfying presented methodology. It shows clearly how to take advantage of the method to treat large examples. They clearly show the interest of this new method to treat large examples.Originality/valueThe paper demonstrates clearly that verification of large finite element problem must have dedicated methods in order to be applicable.

Recent progress in computer hardware technology and computational mechanics enables us to perform large scale finite element (FE) analyses which can simulate physical phenomena more precisely. Because of slow technology propagation from a research stage to a development stage of FE analysis programs, however, many researchers and engineers can not enjoy benefits of large scale FE analyses. For advanced users in the computational mechanics field who want to implement their own analysis schemes or to modify existing analysis programs for their aim, it is a serious problem.

Development of a simplified plate element for large deflection elasto-plastic finite element analysis

Mechanical testing methods to determine the material constants for large deformation nonlinear finite element analysis were demonstrated for natural rubber. Uniaxial tension, uniaxial compression, equi-biaxial tension and pure shear tests of rubber specimens are performed to achieve the stress-strain curves. The stress-strain curves are obtained after between 5 and 10 cycles to consider the Mullins effect. Mooney and Ogden strain-energy density functions, which are typical form of the hyperelastic material, are determined and compared with each other. The material constants using only uniaxial tension data are about 20% higher than those obtained by any other test data set. The experimental equations of shear elastic modulus on the hardness and maximum strain are presented using multiple regression method. Large deformation finite element analysis of automotive transmission mount using different material constants is performed and the load-displacement curves are compared with experiments. The selection of material constant in large deformation finite element analysis depend on the strain level of component in service.br/

In this paper, one presents, implements and validates a total Lagrangian beam finite element formulation capable of handling large displacements and finite rotations, as well as cross-section in-plane (distortion and local bending) and out-of-plane (warping) deformations. This formulation can be viewed as a generalisation of the well-known Reissner-Simo beam theory that includes warping/transverse bending deformation modes and in which the member walls can undergo in-plane finite relative rotations. When compared with a shell finite element discretisation, the proposed beam formulation has the distinct advantage of drastically reducing the number of degrees-of-freedom required to achieve equally accurate results. Initially, the kinematical description of the member is addressed, devoting special attention to the assessment of the transverse plate bending effects associated with cross-section distortion. Next, the equilibrium equations and the corresponding symmetric tangent operator are obtained, by assuming an additive update of the rotational parameters. In order to illustrate the application, provide validation and show the capabilities of the proposed formulation, several numerical results are presented, discussed and compared with values yielded by large displacement shell finite element analyses (see Fig. 1) - one obtains a good correlation in all cases, which clearly demonstrates the vast potential of the proposed formulation. Open image in new window Figure 1 Beam deformed configurations yielded by the (a) proposed beam formulation and (b) shell element model

A three-dimensional, multiple-well, numerical simulator for simulating single- or two-phase flow of water and oil is developed for fractured reservoirs. The simulator equations are two-phase flow extensions of the single-phase flow equations derived by Warren and Root. The simulator accounts for relative fluid mobilities, gravity force, imbibition, and variation in reservoir properties. The simulator handles uniformly and nonuniformly properties. The simulator handles uniformly and nonuniformly distributed fractures and for no fractures at all. The simulator can be used to simulate the water-oil displacement process and in the transient testing of fractured reservoirs. The simulator was used on the conceptual models of two naturally fractured reservoirs: a quadrant of a five-spot reservoir and a live-well dipping reservoir with water drive. These results show the significance of imbibition in recovering oil from the reservoir rock in reservoirs with an interconnected fracture network. Introduction Numerical reservoir simulators are being used extensively to simulate multiphase, multicomponent flow in "single-porosity" petroleum reservoirs. Such simulators generally cannot be used to petroleum reservoirs. Such simulators generally cannot be used to study flow behavior in the naturally fractured reservoirs that are usually classified as double-porosity systems. In the latter, one porosity is associated with the matrix blocks and the other porosity is associated with the matrix blocks and the other represents that of the fractures and vugs. If fractures provide the main path for fluid flow from the reservoir, then usually the oil from the matrix blocks flows into the fracture space, and the fractures carry the oil to the wellbore. When water comes in contact with the oil zone, water may imbibe into the matrix blocks to displace oil. Combinations of large flow rates, low matrix permeability, and weak imbibition may result in water fingering permeability, and weak imbibition may result in water fingering through the fractures into the wellbore. Once fingering of water occurs, the water-oil ratio may increase to a large value. None of the published theoretical work on multiphase flow in naturally fractured systems has been applied directly to the simulation of a reservoir as a whole. Usually, only a segment of the reservoir was simulated, and the results were extrapolated to the entire reservoir. To simulate a reservoir as a whole, we have developed a mathematical formulation of the flow problem that has been programmed as a three-dimensional, compressible, water-oil reservoir simulator. The simulator equations are two-phase flow extensions of the single-phase flow equations derived by Warren and Root. The theory is based on the assumption of double porosity at each point in a manner that the fractures form a continuum filled by the noncontinuous matrix blocks. In other words, the fractures are the boundaries of the matrix blocks. The flow equations are solved by a finite difference method. A typical finite-difference grid cell usually contains one or several matrix blocks. In this case, all the matrix blocks within the finite-difference grid cell have the same pressure and saturation. Gravity segregation within individual matrix blocks is not calculated, but the over-all gravity segregation from one grid cell to another is accounted for. In many practical problems, this approximation is acceptable. In some situations, a matrix block encloses several finite-difference grid cells. In this case, the gravity segregation within the matrix block is calculated. To include heterogeneity, a redefinition of local porosities and permeabilities provides a method for simulating situations where part of the reservoir is fractured and where part is not fractured. The above description points to the complexity of the situations that one encounters. Therefore, the judicious choice of the number of finite difference grid cells with respect to the number of matrix blocks becomes a critical engineering decision. Later sections will provide insight to alleviate such decisions. SPEJ P. 317

Incomplete factorization has been shown to be a good preconditioner for the conjugate gradient method on a wide variety of problems. It is well known that allowing some fill-in during the incomplete factorization can significantly reduce the number of iterations needed for convergence. Allowing fill-in, however, increases the time for the factorization and for the triangular system solutions. Additionally, it is difficult to predict a priori how much fill-in to allow and how to allow it. The unpredictability of the required storage/work and the unknown benefits of the additional fill-in make such strategies impractical to use in many situations. In this article we motivate, and then present, two “black-box” strategies that significantly increase the effectiveness of incomplete Cholesky factorization as a preconditioner. These strategies require no parameters from the user and do not increase the cost of the triangular system solutions. Efficient implementations for these algorithms are described. These algorithms are shown to be successful for a variety of problems from the Harwell-Boeing sparse matrix collection.

Block SSOR preconditionings for high-order 3D FE systems. II. Incomplete BSSOR preconditionings

A class of Generalized Approximate Inverse Matrix (GAIM) techniques, based on the concept of LU-sparse factorization procedures, is introduced for computing explicitly approximate inverses of large sparse unsymmetric matrices of irregular structure, without inverting the decomposition factors. Explicit preconditioned iterative methods, in conjunction with modified forms of the GAIM techniques, are presented for solving numerically initial/boundary value problems on multiprocessor systems. Application of the new methods on linear boundary-value problems is discussed and numerical results are given.

This paper presents a large strain finite element formulation based on strain-gradient plasticity theory which is suitable for simulation of strain localization. A local elasto-plastic constitutive relation introduced in the formulation is characterized by the independence from mesh discretization. A mixed formulation with finite element and Lagrange multipliers is adopted, which enables 2D and 3D formulation without any difficulties similar to the ones of the C1 finite elements. 1D model is formulated at first and is fully examined by comparing the analytical solutions with an earlier model with Hermitian elements (Chambon et al. 1998). A 2D model is then formulated straightforwardly and a series of biaxial test are performed to investigate a 2D shear-banding problem.

Finite element simulations of the geometrically nonlinear, large angle motion of flexible bodies are presented and discussed. The simulations use ABAQUS, a commercially available nonlinear, large strain-displacement structural finite element program. To evaluate the capabilities of ABAQUS, simulation results for three examples are presented: a large angle maneuver of a cantilevered beam, the nutation of a rotating free-free beam observed from a rotating frame, and the spin-up of a cantilevered beam from rest. The spin-up problem is a standard test case which tests for false divergence of the beam above a critical spin rate. In these simple examples, ABAQUS successfully simulates the large rotational motion, correctly capturing the geometrically nonlinear effects. The success of these simple examples indicates that ABAQUS, or similar nonlinear finite element programs, could be used to simulate the large angle rotation of arbitrary structures when a high fidelity t ruth model is desired.

Large scale structural synthesis