Shape derivatives for an augmented Lagrangian formulation of elastic contact problems

This work deals with shape optimization of an elastic body in sliding contact (Signorini) with a rigid foundation. The mechanical problem is written under its augmented Lagrangian formulation, then solved using a classical iterative approach. For practical reasons we are interested in applying the optimization process with respect to an intermediate solution produced by the iterative method. Due to the projection operator involved at each iteration, the iterate solution is not classically shape differentiable. However, using an approach based on directional derivatives, we are able to prove that it is conically differentiable with respect to the shape, and express sufficient conditions for shape differentiability. Finally, from the analysis of the sequence of conical shape derivatives of the iterative process, conditions are established for the convergence to the conical derivative of the original contact problem.

Numerical Modeling Methods for Large Open Loop Multibody System

A pendulum’s motion was stated to be as a way to illustrate the movement of human body in the studies of multibody system. Therefore, a comparison between the two numerical models in multibody systems were implemented on the articulated pendulums of different sizes. The two numerical models were known as the augmented Lagrangian formulation and fully recursive method. In order to identify the difference performance of the numerical models, various size of articulated pendulums has been tested which are 2, 4, 8, 16, 20 and 40 pendulums. Differential equations developed from both models were solved by using Runge-Kutta 4 and 5. Both models were coded in Matlab and have been optimized in order to ensure only related routine were considered. The performance was evaluated based on the computing time with constant relative and absolute tolerance in Runge-Kutta solver which is 0.01 s. All pendulums were assumed to have the same weight, angle and length. As for the results, the augmented Lagrangian formulation solved the differential equations faster than the fully recursive method when tested up to 20 pendulums. However, fully recursive method started to solve the differential equations faster than the augmented Lagrangian method when it need to deal with a very large system such as 40 pendulums and above. Thus, it can be concluded that the suitable method to solve the small, open loop system such as articulated pendulums is augmented Lagrangian method while for a very large system, the fully recursive method will be more efficient.

Quasi-optimal and pressure-robust discretizations of the Stokes equations by new augmented Lagrangian formulations

We approximate the solution of the stationary Stokes equations with various conforming and nonconforming inf-sup stable pairs of finite element spaces on simplicial meshes. Based on each pair, we design a discretization that is quasi-optimal and pressure-robust, in the sense that the velocity $H^1$-error is proportional to the best velocity $H^1$-error. This shows that such a property can be achieved without using conforming and divergence-free pairs. We also bound the pressure $L^2$-error, only in terms of the best velocity $H^1$-error and the best pressure $L^2$-error. Our construction can be summarized as follows. First, a linear operator acts on discrete velocity test functions, before the application of the load functional, and maps the discrete kernel into the analytical one. Second, in order to enforce consistency, we employ a new augmented Lagrangian formulation, inspired by discontinuous Galerkin methods.