YU Hang

Polyhedral projection is a main operation of the polyhedron abstract domain. It can be computed via parametric linear programming (PLP), which is more efficient than the classic Fourier-Motzkin elimination method. In prior work, PLP was done in arbitrary precision rational arithmetic. In this paper, we present an approach where most of the computation is performed in floating-point arithmetic, then exact rational results are reconstructed. We also propose a workaround for a difficulty that plagued previous attempts at using PLP for computations on polyhedra: in general the linear programming problems are degenerate, resulting in redundant computations and geometric descriptions.

In product or system design, deterministic optimization approaches are widely used nowadays. However, these approaches do not take into account the uncertainties inherent to the models used, which can sometimes lead to unreliable solutions. It is then appropriate to focus on stochastic optimization approaches. Reliablity Based Robust Design Optimization (RBRDO) approaches take into account the uncertainties during the optimization through an additional uncertainty analysis loop (Uncertainty Anlysis, UA). For most practical applications, UA is performed by Monte Carlo Simulation (MCS) combined with structural analysis. The major disadvantage of this type of approach is the computational cost which is prohibitive. Therefore, we are interested in developing efficient methodologies for the implementation of RBRDOs based on MCS analysis. We present a UA method based on MCS analysis in which the random response is approximated on a Polynomial Chaos Expansion (PCE) basis. Thus, the efficiency of UA is greatly improved by avoiding too much repetition of structural analyses. Unfortunately, this approach is not relevant for high-dimensional problems, for example for applications in dynamics. We therefore propose to approximate the dynamic response by taking into account only the resolution to random eigenvalues. In this way, only the random structural parameters appear in the PCE. To deal with the problem of mode mixing in our approach, we have relied on the MAC factor which allows to quantify it. We have developed a univariate method to check which variable generates mode mixing in order to reduce or eliminate it. Next, we present a sequential RBRDO approach to improve the efficiency and avoid the non-convergence problems present in RBRDO approaches. In our approach, we extended the classical sequential strategy, mainly aiming at decoupling the reliability analysis from the optimization procedure, by separating the moment evaluation from the optimization loop. We used a local exponential approximation around the current design point to construct equivalent deterministic objectives and stochastic constraints. In order to obtain the different coefficients for our approximation, we have developed a robustness sensitivity analysis based on an auxiliary distribution as well as a moment sensitivity analysis based on the PCE approach. We show the relevance and the efficiency of the proposed approaches through different numerical examples. We then apply our RBRDO approach to the design of a damper in the field of passive vibration control of a structure with random quantities. The results obtained by our approach allow not only to reduce the variability of the response, but also to better control the amplitude of the response through a threshold chosen in advance.