The dissertation begins by treating the problem of frictionless quasi-static contact from an analytical and numerical standpoint. The strong and weak forms of the problem are derived from basic principles, and a number of key points are discussed. The criteria of the patch test and the Ladyszhenskaya-Babus˘ka-Brezzi condition1 are applied to both the Signorini 2 and two-body contact problem. The satisfaction of these two criteria is shown to be critical to the success of a contact algorithm. The LBB condition is a requirement on the construction of numerical methods for problems with constraints using two fields, a field of unconstrained primary variables, and a second field of Lagrange multipliers which enforce the constraints. In order to satisfy the condition, the Lagrange multiplier field has to be constructed without over-constraining the primary variable, or stability problems may occur. Note that the application of the LBB condition to the two-body problem is a new addition to the literature. This machinery is used to analyze existing contact algorithms. It is shown that existing geometrically-unbiased contact algorithms for the two-body problem fail one or both of these criteria.; A novel geometrically unbiased two-pass reduced-constraint scheme is presented. It is shown that this method unequivocally passes both the patch test and the LBB condition.; The dissertation proceeds on with a discussion of frictionless dynamic contact. The analytical background is reviewed, including a number of exact and approximate analytical results. Contact/impact problems modeled via the semi-discrete finite element method are shown to exhibit an instability. This instability is shown to be due to the effect of the discretization, which prohibits the formation of shock waves.; A novel technique for modifying a time integration scheme to remove this instability is presented. This technique is motivated by a novel treatment of the problem from the standpoint of the dynamics of continuous Hamiltonian systems. The technique is shown to be generally applicable to arbitrary contact schemes and time integration methods. It is demonstrated that the technique removes the instability without requiring a globally-dissipative numerical scheme. (Abstract shortened by UMI.); 1The Ladyszhertskaya-Babus˘ka-Brezzi condition is oftentimes referred to by the shorter term Babus˘ka-Brezzi condition, or abbreviated as the LBB condition. 2The Signorini problem is the general problem of contact between a deformable body and a rigid obstacle, which was first rigorously formulated by Signorini [Sig33, Sig59].
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