Stochastic optimization of nonlinear energy sinks

Nonlinear energy sinks (NES) are a promising technique to achieve vibration mitigation. Through nonlinear stiffness properties, NES are able to passively and irreversibly absorb energy. Unlike the traditional Tuned Mass Damper (TMD), NES absorb energy from a wide range of frequencies. Many studies have focused on NES behavior and dynamics, but few have addressed the optimal design of NES. Design considerations of NES are of prime importance as it has been shown that NES dynamics exhibit an acute sensitivity to uncertainties. In fact, the sensitivity is so marked that NES efficiency is near-discontinuous and can switch from a high to a low value for a small perturbation in design parameters or loading conditions. This article presents an approach for the probabilistic design of NES which accounts for random design and aleatory variables as well as response discontinuities. In order to maximize the mean efficiency, the algorithm is based on the identification of regions of the design and aleatory space corresponding to markedly different NES efficiencies. This is done through a sequence of approximated sub-problems constructed from clustering, Kriging approximations, a support vector machine, and Monte-Carlo simulations. The refinement of the surrogates is performed locally using a generalized max-min sampling scheme which accounts for the distributions of random variables. The sampling scheme also makes use of the predicted variance of the Kriging surrogates for the selection of aleatory variables values. The proposed algorithm is applied to three example problems of varying dimensionality, all including an aleatory excitation applied to the main system. The stochastic optima are compared to NES optimized deterministically.     

Controlling structural failure modes during an impact in the presence of uncertainties

A methodology to enforce a given structural dynamic behavior during an impact while accounting for uncertainty is presented. The approach is based on locating structural fuses that weaken the structure locally and help enforce a deformation mode. The problem of enforcing the crushing of a tube impacting a rigid wall is chosen. In order to find the positions of the fuses, the method identifies distinct structural dynamic behaviors using designs of experiments and clustering techniques. The changes in behavior are studied with respect to variations of the fuse positions and random parameters, such as the thickness. Based on the probabilistic distributions, a measure of the likelihood of occurrence of global buckling is defined. The positions of the fuses are defined using an optimization problem in terms of the likelihood of global buckling and the amount of absorbed energy in the tube. A first formulation of the problem considers variability in the tube’s thickness only. A second formulation also accounts for uncertainties in the positions of the fuses.     

A multifidelity approach for the construction of explicit decision boundaries: application to aeroelasticity

This paper presents a multifidelity approach for the construction of explicit decision boundaries (constraints or limit-state functions) using support vector machines. A lower fidelity model is used to select specific samples to construct the decision boundary corresponding to a higher fidelity model. This selection is based on two schemes. The first scheme selects samples within an envelope constructed from the lower fidelity model. The second technique is based on the detection of regions of inconsistencies between the lower and the higher fidelity decision boundaries. The approach is applied to analytical examples as well as an aeroelasticity problem for the construction of a nonlinear flutter boundary.     

A displacement based optimization method for geometrically nonlinear frame structures

An extension of the displacement based optimization method to frames with geometrically nonlinear response is presented. This method, when applied to small-scale trusses with linear and nonlinear response, appeared to be efficient providing the same solutions as the classical optimization method. The efficiency of the method is due to the elimination of numerous finite element analyses that are required in using the traditional optimization approach. However, as opposed to trusses, frame problems have typically a larger number of degrees of freedom than cross sectional area design variables. This leads to difficulties in the implementation of the method compared to the truss implementation. A scheme that relaxes the nodal equilibrium equations is introduced, and the method is validated using test examples. The optimal designs obtained by using the displacement based optimization and the classical approaches are compared to validate the application to frame structures. The characteristics and limitations of the optimization in the displacement space for sizing problems, based on the current formulation, are discussed.     

Laboratory study on the liquefaction resistance of sand-silt mixtures: effect of grading characteristics

The liquefaction susceptibility of saturated medium sand-silt mixture samples is evaluated by monotonic and cyclic undrained triaxial laboratory tests that were carried out on reconstituted specimens at various relative densities (D_r   =  20, 53 and 91%) and a constant confining pressure (s₃^￠=100{\sigma_{3}^{\prime}=100}  kPa). The test results were used to conclude on the effect of grading characteristics and other parameters on the liquefaction resistance of the sand-silt mixtures. The monotonic test results indicate that the undrained shear strength at the peak and the undrained residual strength can be correlated to the coefficient of uniformity (C_u) and the average diameter (D₅₀). Indeed, they decrease linearly with the increase of the uniformity coefficient and decrease of the average diameter. It is found that a relationship between the liquefaction resistance and any of the diameters (D₁₀ or D₅₀) and the coefficient of uniformity (C_u) would be more realistic than to build a relation between the coefficient of gradation (C_c) and the liquefaction resistance. Undrained cyclic triaxial tests indicate that the cyclic liquefaction resistance of the sand-silt mixtures decreases linearly with the decrease of the effective diameter (D₁₀) and mean size (D₅₀) and increase of the fines content for the fines content range tested (Fc  =  0–40%).     

A generalized “max-min” sample for surrogate update

The preliminary Multidisciplinary Design and Optimisation of a flexible wing aerofoil apropos a small Unmanned Air Vehicle is performed using a multifidelity model-based strategy. Both the passively adaptive structure and the shape of the flexible wing aerofoil are optimised for best aerodynamic performance under aero-structural constraints, within a coupled aeroelastic formulation. A typical flight mission for surveillance purposes is considered and includes the potential occurrence of wind gusts. A metamodel for the high-fidelity objective function and each of the constraints is built, based on a tuned low-fidelity one, in order to improve the efficiency of the optimisation process. Both metamodels are based on solutions of the aeroelastic equations for a flexible aerofoil but employ different levels of complexity and computational cost for modelling aerodynamics and structural dynamics within a modal approach. The high-fidelity model employs nonlinear Computational Fluid Dynamics coupled with a full set of structural modes, whereas the low-fidelity one employs linear thin aerofoil theory coupled with a reduced set of structural modes. The low-fidelity responses are then corrected according to few high-fidelity responses, as prescribed by an appropriate Design of Experiment, by means of a suitable tuning technique. A standard Genetic Algorithm is hence utilised to find the global optimum, showing that a flexible aerofoil is characterised by higher aerodynamic efficiency than its rigid counterpart. Wing weight reduction is also accomplished when a Multiobjective Genetic Algorithm is adopted.     

Multi-level A Priori Hyper-Reduction of mechanical models involving internal variables

This paper concerns the adaptation of reduced-order models during simulations of series of elastoviscoplastic problems. In continuation with previous works, this paper aimed at extending the A Priori Hyper-Reduction method (APHR method) for nonlinear thermal problems to nonlinear mechanical problems involving internal variables. This method is an a priori approach because full incremental responses of detailed models are not forecasted in order to build reduced-order models. The recent extension of the Hyper-Reduction method to reduction of mechanical models involving internal variables makes possible the reduction of degrees of freedom and the reduction of integration points. A multi-level formulation is introduced to focus on the capability of the method to perform efficient parallel computations to adapt reduced-order models.     

A robust adaptive model reduction method for damage simulations

In our opinion, many of complex numerical models in materials science can be reduced without losing their physical sense. Due to solution bifurcation and strain localization of continuum damage problems, damage predictions are very sensitive to any model modification. Most of the robust numerical algorithms intend to forecast one approximate solution of the continuous model despite there are multiple solutions. Some model perturbations can possibly be added to the finite element model to guide the simulation toward one of the solutions. Doing a model reduction of a finite element damage model is a kind of model perturbation. If no quality control is performed the prediction of the reduced-order model (ROM) can really differ from the prediction of the full finite element model. This can happen using the snapshot Proper Orthogonal Decomposition (POD) model reduction method. Therefore, if the expected purpose of the reduced approximation is to estimate the solution that the finite element simulation should give, an adaptive reduced-order modeling is required when reducing finite element damage models.We propose an adaptive reduced-order modeling method that enables to estimate the effect of loading modifications. The Rousselier continuum damage model is considered. The differences between the finite element prediction and the one provided by the adapted reduced-order model (ROM) remain stable although various loading perturbations are introduced. The adaptive algorithm is based on the APHR (A Priori Hyper Reduction) method. This is an incremental scheme using a ROM to forecast an initial guess solution to the finite element equations. If, at the end of a time increment, this initial prediction is not accurate enough, a finite element correction is added to the ROM prediction. The proposed algorithm can be viewed as a two step Newton–Raphson algorithm. During the first step the prediction belongs to the functional space related to the ROM and during the second step the correction belongs to the classical FE functional space. Moreover the corrections of the ROM predictions enable to expand the basis related to the ROM. Therefore the ROM basis can be improved at each increment of the simulation. The efficiency of the adaptive algorithm is checked comparing the amount of global linear solutions involved in the proposed scheme versus the amount of global linear solutions involved in the classical incremental Newton–Raphson scheme. The quality of the proposed approximation is compared to the one provided by the classical snapshot Proper Orthogonal Decomposition (POD) method.     

Study of a new local update scheme for cellular automata in structural design

This paper investigates an improved local update scheme for cellular automata (CA) applied to structural design. Local analysis and design rules are derived for equilibrium and minimum compliance design. The new update scheme consists of repeating analysis and optimality-based design rules locally. The benefits of this approach are demonstrated through a series of systematic experiments. Truss topology design problems of various sizes are used based on the Gauss–Seidel and the Jacobi iteration modes. Experiments show the robust convergence of the approach as compared to an earlier CA implementation. The approach is also extended to a plate problem.     

Constrained efficient global optimization with support vector machines

This paper presents a methodology for constrained efficient global optimization (EGO) using support vector machines (SVMs). While the objective function is approximated using Kriging, as in the original EGO formulation, the boundary of the feasible domain is approximated explicitly as a function of the design variables using an SVM. Because SVM is a classification approach and does not involve response approximations, this approach alleviates issues due to discontinuous or binary responses. More importantly, several constraints, even correlated, can be represented using one unique SVM, thus considerably simplifying constrained problems. In order to account for constraints, this paper introduces an SVM-based ??probability of feasibility?? using a new Probabilistic SVM model. The proposed optimization scheme is constituted of two levels. In a first stage, a global search for the optimal solution is performed based on the ??expected improvement?? of the objective function and the probability of feasibility. In a second stage, the SVM boundary is locally refined using an adaptive sampling scheme. An unconstrained and a constrained formulation of the optimization problem are presented and compared. Several analytical examples are used to test the formulations. In particular, a problem with 99 constraints and an aeroelasticity problem with binary output are presented. Overall, the results indicate that the constrained formulation is more robust and efficient.