Clark R. Dohrmann

A Preconditioner for Substructuring Based on Constrained Energy Minimization

A preconditioner for substructuring based on constrained energy minimization concepts is presented. The preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second- and fourth-order structural mechanics problems. The approach involves constraints associated with disjoint sets of nodes on substructure boundaries. These constraints provide the means for preconditioning at both the substructure and global levels. Numerical examples are presented that demonstrate the good performance of the method in terms of iterations, compute time, and condition numbers of the preconditioned equations.

Node‐based uniform strain elements for three‐node triangular and four‐node tetrahedral meshes

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.

Convergence of a balancing domain decomposition by constraints and energy minimization

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 corners 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+log2(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.

Control for slosh-free motion of an open container

This article describes two methods for controlling the surface of a liquid in an open container as it is being carried by a robot arm. Both methods make use of the fundamental mode of oscillation and damping of the liquid in the container as predicted from a boundary element model of the fluid. The first method uses an infinite impulse response filter to alter the acceleration profile so that the liquid remains level except for a single wave at the beginning and end of the motion. The motion of the liquid is similar to that of a simple pendulum. The second method removes the remaining two surface oscillations by tilting the container parallel to the beginning and ending wave. A double pendulum model is used to determine the trajectory for this motion. Experimental results of a FANUC S-800 robot moving a 230 mm diameter hemispherical container of water are presented.

Methods for connecting dissimilar three-dimensional finite element meshes†

Two methods are presented for connecting dissimilar three-dimensional finite element meshes. The first method combines the concept of master and slave surfaces with the uniform strain approach for finite elements. By modifying the boundaries of elements on a slave surface, corrections are made to element formulations such that first-order patch tests are passed. The second method is based entirely on constraint equations, but only passes a weaker form of the patch test for non-planar surfaces. Both methods can be used to connect meshes with 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. Published in 2000 by John Wiley & Sons, Ltd.

Flexible Robot Dynamics and Controls

Contents. 1. Introduction. 2. Mathematical Preliminaries. 3. Flexible Robot Dynamic Modeling. 4. System Identification. 5. Input Shaping for Path Planning. 6. Linear Feedback Control. 7. Nonlinear Systems and Sliding Mode Control. 8. Adaptive Sliding Mode Control. Appendix A: VF02AD Optimization. Appendix B: MATLAB%% Optimization. Appendix C: Hardware & Software Support.

An Overlapping Schwarz Algorithm for Almost Incompressible Elasticity

Overlapping Schwarz methods are extended to mixed finite element approximations of linear elasticity which use discontinuous pressure spaces. The coarse component of the preconditioner is based on a low-dimensional space previously developed for scalar elliptic problems and a domain decomposition method of iterative substructuring type, i.e., a method based on nonoverlapping decompositions of the domain, while the local components of the preconditioner are based on solvers on a set of overlapping subdomains. A bound is established for the condition number of the algorithm which grows in proportion to the logarithm of the number of degrees of freedom in individual subdomains and, essentially, to the third power of the relative overlap between the overlapping subdomains, and which is independent of the Poisson ratio as well as jumps in the Lame parameters across the interface between the subdomains. A positive definite reformulation of the discrete problem makes the use of the standard preconditioned conjugate gradient method straightforward. Numerical results, which include a comparison with problems of compressible elasticity, illustrate the findings.

Domain Decomposition for Less Regular Subdomains: Overlapping Schwarz in Two Dimensions

In the theory of domain decomposition methods, it is often assumed that each subdomain is the union of a small set of coarse triangles or tetrahedra. In this study, extensions to the existing theory which accommodate subdomains with much less regular shapes are presented; the subdomains are required only to be John domains. Attention is focused on overlapping Schwarz preconditioners for problems in two dimensions with a coarse space component of the preconditioner, which allows for good results even for coefficients which vary considerably. It is shown that the condition number of the domain decomposition method is bounded by

A method for connecting dissimilar finite element meshes in two dimensions

C(1+H/\delta)(1+\log(H/h))^2

An approximate BDDC preconditioner

, where the constant C is independent of the number of subdomains and possible jumps in coefficients between subdomains. Numerical examples are provided which confirm the theory and demonstrate very good performance of the method for a variety of subregions including those obtained when a mesh partitioner is used for the domain decomposition.