Publications

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The focus of this paper is a penalty-based strategy for preconditioning elliptic saddle point systems. As the starting point, we consider the regularization approach of Axelsson in which a related linear system, differing only in the (2,2) block of the coefficient matrix, is introduced. By choosing this block to be negative definite, the dual unknowns of the related system can be eliminated resulting in a positive definite primal Schur complement. Rather than solving the Schur complement system exactly, an approximate solution is obtained using a substructuring preconditioner. The approximate primal solution together with the recovered dual solution then define the preconditioned residual for the original system.

We present a new family of stabilized methods for the Stokes problem. The focus of the paper is on the lowest order velocity-pressure pairs. While not LBB compliant, their simplicity and attractive computational properties make these pairs a popular choice in engineering practice. Our stabilization approach is motivated by terms that characterize the LBB "deficiency" of the unstable spaces. The stabilized methods are defined by using these terms to modify the saddle-point Lagrangian associated with the Stokes equations. The new stabilized methods offer a number of attractive computational properties. In contrast to other stabilization procedures, they are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems. Furthermore, the new methods are unconditionally stable, achieve optimal accuracy with respect to solution regularity, and have simple and straightforward implementations. We present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods. © 2006 Society for Industrial and Applied Mathematics.

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The focus of this paper is a penalty-based strategy for preconditioning elliptic saddle point systems. As the starting point, we consider the regularization approach of Axelsson in which a related linear system, differing only in the (2,2) block of the coefficient matrix, is introduced. By choosing this block to be negative definite, the dual unknowns of the related system can be eliminated resulting in a positive definite primal Schur complement. Rather than solving the Schur complement system exactly, an approximate solution is obtained using a substructuring preconditioner. The approximate primal solution together with the recovered dual solution then define the preconditioned residual for the original system. The approach can be applied to a variety of different saddle point problems. Although the preconditioner itself is symmetric and indefinite, all the eigenvalues of the preconditioned system are real and positive if certain conditions hold. Stronger conditions also ensure that the eigenvalues are bounded independently of mesh parameters. An interesting feature of the approach is that conjugate gradients can be used as the iterative solution method rather than GMRES. The effectiveness of the overall strategy hinges on the preconditioner for the primal Schur complement. Interestingly, the primary condition ensuring real and positive eigenvalues is satisfied automatically in certain instances if a Balancing Domain Decomposition by Constraints (BDDC) preconditioner is used. Following an overview of BDDC, we show how its constraints can be chosen to ensure insensitivity to parameter choices in the (2,2) block for problems with a divergence constraint. Examples for different saddle point problems are presented and comparisons made with other approaches.

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Large-scale finite element analysis often requires the iterative solution of equations with many unknowns. Preconditioners based on domain decomposition concepts have proven effective at accelerating the convergence of iterative methods like conjugate gradients for such problems. A study of two new domain decomposition preconditioners is presented here. The first is based on a substructuring approach and can viewed as a primal counterpart of the dual-primal variant of the finite element tearing and interconnecting method called FETI-DP. The second uses an algebraic approach to construct a coarse problem for a classic overlapping Schwarz method. The numerical properties of both preconditioners are shown to scale well with problem size. Although developed primarily for structural mechanics applications, the preconditioners are also useful for other problems types. Detailed descriptions of the two preconditioners along with numerical results are included.

A convergence theory is presented for a substructuring preconditioner based on constrained energy minimization concepts. The substructure spaces consist of local functions with zero values of the constraints, while the coarse space consists of minimal energy functions with the constraint values continuous across substructure interfaces. In applications, the constraints include values at comers and optionally averages on edges and faces. The preconditioner is reformulated as an additive Schwarz method and analysed by building on existing results for balancing domain decomposition. The main result is a bound on the condition number based on inequalities involving the matrices of the preconditioner. Estimates of the form C(1 + log 2(H/h)) are obtained under the standard assumptions of substructuring theory. Computational results demonstrating the performance of method are included. Published in 2003 by John Wiley & Sons, Ltd.

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The ability to generate a suitable finite element mesh in an automatic fashion is becoming the key to being able to automate the entire engineering analysis process. However, placing an all-hexahedron mesh in a general three-dimensional body continues to be an elusive goal. The approach investigated in this research is fundamentally different from any other that is known of by the authors. A physical analogy viewpoint is used to formulate the actual meshing problem which constructs a global mathematical description of the problem. The analogy used was that of minimizing the electrical potential of a system charged particles within a charged domain. The particles in the presented analogy represent duals to mesh elements (i.e., quads or hexes). Particle movement is governed by a mathematical functional which accounts for inter-particles repulsive, attractive and alignment forces. This functional is minimized to find the optimal location and orientation of each particle. After the particles are connected a mesh can be easily resolved. The mathematical description for this problem is as easy to formulate in three-dimensions as it is in two- or one-dimensions. The meshing algorithm was developed within CoMeT. It can solve the two-dimensional meshing problem for convex and concave geometries in a purely automated fashion. Investigation of the robustness of the technique has shown a success rate of approximately 99% for the two-dimensional geometries tested. Run times to mesh a 100 element complex geometry were typically in the 10 minute range. Efficiency of the technique is still an issue that needs to be addressed. Performance is an issue that is critical for most engineers generating meshes. It was not for this project. The primary focus of this work was to investigate and evaluate a meshing algorithm/philosophy with efficiency issues being secondary. The algorithm was also extended to mesh three-dimensional geometries. Unfortunately, only simple geometries were tested before this project ended. The primary complexity in the extension was in the connectivity problem formulation. Defining all of the interparticle interactions that occur in three-dimensions and expressing them in mathematical relationships is very difficult.

Structural dynamic systems are often attached to a support structure to simulate proper boundary conditions during testing. In some cases, the support structure is fairly simple and can be modeled by discrete springs and dampers. In other cases, the desired test conditions necessitate the use of a support structure which introduces dynamics of its own. For such cases, a more complex structural dynamic model is required to simulate the response of the full combined system. In this paper, experimental frequency response functions, admittance function modeling concepts, and least squares reductions are used to develop a support structure model including both translational and rotational degrees of freedom at an attachment location. Subsequently, the modes of the support structure are estimated, and a NASTRAN model is created for attachment to the tested system.

A family of uniform strain elements is presented for three-node triangular and four-node tetrahedral meshes. The elements use the linear interpolation functions of the original mesh, but each element is associated with a single node. As a result, a favorable constraint ratio for the volumetric response is obtained for problems in solid mechanics. The uniform strain elements do not require the introduction of additional degrees of freedom and their performance is shown to be significantly better than that of three-node triangular or four-node tetrahedral elements. In addition, nodes inside the boundary of the mesh are observed to exhibit superconvergent behavior for a set of example problems.

This report documents a collection of papers on a family of uniform strain tetrahedral finite elements and their connection to different element types. Also included in the report are two papers which address the general problem of connecting dissimilar meshes in two and three dimensions. Much of the work presented here was motivated by the development of the tetrahedral element described in the report "A Suitable Low-Order, Eight-Node Tetrahedral Finite Element For Solids," by S. W. Key {ital et al.}, SAND98-0756, March 1998. Two basic issues addressed by the papers are: (1) the performance of alternative tetrahedral elements with uniform strain and enhanced uniform strain formulations, and (2) the proper connection of tetrahedral and other element types when two meshes are "tied" together to represent a single continuous domain.

A method is presented for connecting dissimilar finite element meshes in three dimensions. The method combines the concept of master and slave surfaces with the uniform strain approach for surface, corrections finite elements- By modifyhg the are made to element formulations boundaries of elements on the slave such that first-order patch tests are passed. The method can be used to connect meshes which use different element types. In addition, master and slave surfaces can be designated independently of relative mesh resolutions. Example problems in three-dimensional linear elasticity are presented.

A method is presented for estimating uncertain or unknown parameters in a mathematical model using measurements of transient response. The method is based on a least squares formulation in which the differences between the model and test-based responses are minimized. An application of the method is presented for a nonlinear structural dynamic system. The method is also applied to a model of the Department of Energy armored tractor trailer. For the subject problem, the transient response was generated by driving the vehicle over a bump of prescribed shape and size. Results from the analysis and inspection of the test data revealed that a linear model of the vehicle's suspension is not adequate to accurately predict the response caused by the bump.