Shape And Topology Optimization Methods For Elastic Contact Problems

Structural shape and topology optimization techniques are effective approaches for the achievement of high-performance and lightweight design of high-end equipment such as aeronautical and aerospace vehicles.Nevertheless,existing research works of structural optimization are mainly focused on one single component,while a great number of engineering structures are actually assembled from various parts or components,which leads to the ubiquitous mechanical contact phenomenon.It is thus very important to consider contact interactions that notably affect structural performance in structural design and optimization.However,the high non-linearity and non-smoothness of the contact problem make the structural optimization associated with it a very challenging research topic.Most of the current researches concerning contact problems are confined to validation analysis,while structural optimization of contact problems is the technical vulnerability in structure designs.There are still many inadequacies that there lack effective approaches to handle the leakproofness constraint,that the optimized topological configurations produced by the density-based topology optimization methods contain elements with intermediate densities,and that the influencing mechanisms of the friction behavior and the uniformity of contact pressures on topology optimization results are still not clear,etc.Based on the above considerations,developing structural optimization methods for contact problems and exploring the influencing mechanism of the contact effects on structural optimization results are of great significance to provide applicable instructive design principles for engineering practices.Devoted to shape and topology optimization methods for elastic contact problems,the dissertation mainly covers four aspects,that is,shape optimization of contact problems with the leakproofness constraint,topology optimization of frictionless contact problems,topology optimization of frictional contact problems,and topology optimization for the uniformity of contact pressures.The main research works and contributions are summarized as follows.(1)Shape optimization of contact problems is studied for structural design with the requirement of leakproofness.A linear relaxation model is developed to avoid the zero sensitivities accompanying zero contact pressures caused by the contact non-linearity,thereby resolving the difficulty in the satisfaction of leakproofness constraints using gradient-based algorithms.By introducing a relaxation parameter,the zero contact force is modified to the product of the contact gap and the relaxation parameter.Accordingly,non-zero sensitivities with right searching directions can be obtained to drive gradient-based algorithms to find feasible optimized designs satisfying leakproofness constraints even if they are violated at the initial iteration step.The proposed relaxation model is easy to implement and no modification needs to be done towards the kernel of existing contact analysis code since it is just a postprocessing routine of the finite element analysis results.The effectiveness of the relaxation model and the implemented shape optimization procedure is first validated by analytical examples and then applied successfully to the optimization design of an assembled aero-engine structure for leakproofness.Numerical results show that the relaxation model is robust because it works well with a wide range of relaxation parameter values.(2)A topology optimization procedure for stiffness maximization is established for contact problems within the three-field density-based framework to obtain clear topological configurations without intermediate densities.Two geometric constraints are also incorporated to enforce desired minimum length scales on the optimized solutions in favor of the subsequent identification and reconstruction of structural features as well as the satisfaction of manufacturability requirement.The importance of contact interactions in representing accurately the real working conditions and leading to more reasonable optimized configurations are elucidated through numerical examples and theoretical analyses.This indicates that it is necessary to consider contact interactions even in the conceptual topology optimization design stage.(3)Topology optimization for stiffness maximization of frictional contact problems is accomplished.Two load increment reduction rules are proposed for the increment sensitivity analysis of frictional contact problems.By virtue of these two rules,the load increments that do not influence the final calculated sensitivity values can be identified and directly skipped.The number of adjoint equations that need to be solved is thus decreased.As a result,the computational burden of sensitivity analysis is greatly reduced without loss of accuracy of obtained sensitivities.Both theoretical proofs and numerical verification are presented to validate the proposed load increment reduction rules.The influence of the friction behavior on the topology optimization results is explored through comparative numerical investigations.It is found that the optimized compliance generally decreases with the increase of the friction coefficient because the friction behavior helps to resist tangential deformations at the contact interface.(4)A topology optimization method for the uniformity of contact pressures is presented to maximize structural rigidity while reducing local wear and improving fatigue resistance.The variance function is employed as a quantitative measure of the uniformity of contact pressures.The topology optimization problem is formulated by introducing a constraint of contact pressure variance into the traditional formulation of stiffness maximization.The sensitivity analysis formula of the contact pressure variance is derived by using the adjoint method.Numerical results evidence that the uniformity of contact pressures can be effectively controlled through the constraint of contact pressure variance.But the uniformity of contact pressures is improved at the cost of structural stiffness.Therefore,an appropriate trade-off should be made between the uniformity of contact pressures and the structural stiffness in practical applications according to actual needs.(5)A density threshold method is suggested to deal with the issue of abnormal interruption caused by the numerical instability related to elements with weak materials along the contact boundary,and the issue of incorrect evaluation of contact pressure variance due to the spurious contact phenomenon.With the density threshold method,each contact node pair is assigned with a density according to the physical densities of its adjacent elements.Then,contact node pairs with densities lower than a specific threshold value are identified as void contact node pairs.In the frictional contact topology optimization,all the identified void contact node pairs are prescribed with a zero friction coefficient.As a consequence,the tangential frictional restrictions on elements along the contact boundary are eliminated and the numerical instability is thus avoided.This ensures that the topology optimization process can proceed without any abrupt interruption.In the topology optimization for the uniformity of contact pressures,the identified void contact node pairs are excluded from the set of active contact node pairs to guarantee a correct evaluation of the contact pressure variance.In this way,the optimization process with the consideration of the uniformity of contact pressures is prevented from converging to degenerated solutions.