Lumped mass tensor formulation for dynamic relaxation

The method of dynamic relaxation is an effective means to obtain the steady state solution for finite element problems formulated with explicit time integration. Essentially the method adaptively sets the mass and damping parameters so that the dynamic response is optimally attenuated, leaving only the static solution. Diagonal mass and damping lead to an efficient algorithm, but it has been realized that the inertial properties are not frame invariant, and a monotonic (critically damped) displacement path is not guaranteed. Although this algorithm allows the steady state solution to be computed, it does not ensure no overshoot. Optimally setting the damping parameter can eliminate some overshoots, but in certain cases, overshoot cannot be avoided. This is a serious concern for nonlinear path dependent response. In this paper, a tensor formulation for the mass and damping parameters based on the block diagonal stiffness is presented. This tensor formulation allows all problems to be solved with coordinate frame independence, minimizes overshoot, and provides substantial computational savings by improving the convergence rate.