Gradient-based approximation methods applied to the optimal design of vehicle suspension systems using computational models with severe inherent noise

The principal aim of this paper is to evaluate the feasibility of using gradient-based approximation methods for the optimisation of the spring and damper characteristics of an off-road vehicle, for both ride comfort and handling. The Sequential Quadratic Programming algorithm and the relatively new Dynamic-Q method are the two successive approximation methods used for the optimisation. The determination of the objective function value is performed using computationally expensive numerical simulations that exhibit severe inherent numerical noise. The use of forward finite differences and central finite differences for the determination of the gradients of the objective function within Dynamic-Q is also investigated. This is done in investigating methods for overcoming the difficulties associated with the optimisation of noisy objective functions.A recreational off-road vehicle is modelled in ADAMS, and coupled to MATLAB for the execution of the optimisation process. The full vehicle ADAMS model includes suspension kinematics, a load-dependent tyre model, as well as non-linear springs and dampers. Up to four design variables are considered in modelling the suspension characteristics.It is found that both algorithms perform well in optimising handling. However, difficulties are encountered in obtaining improvements in the design process when ride comfort is considered. Nevertheless, meaningful design configurations are still achievable through the proposed optimisation process, at a relatively low cost in terms of the number of simulations that have to be performed.     

Optimisation of road vehicle passive suspension systems. Part 2. Qualification and case study

As an aid during the concept design phase, the two-dimensional vehicle simulation programme Vehsim2d has been developed (see part 1 of this paper for the vehicle model). The leap-frog optimisation algorithm for constrained problems (LFOPC) has been linked to the multi-body dynamics simulation code (Vehsim2d) to enable the computationally economic optimisation of certain vehicle and suspension design variables. This paper describes the simulation programme, the qualification of the programme, and gives an example of the application of the Vehsim2d/LFOPC system. In particular it is used to optimise the damper characteristics of an existing 22 ton three axle vehicle, over a typical terrain and at a representative speed. By using this system the optimised damper characteristics with respect to ride comfort for the vehicle are computed. The optimum damper characteristics give a 28.5% improvement in the ride comfort of the vehicle over the specified terrain and prescribed speed. Further optimisation runs were performed considering other terrain and different speed values. From these results final damper characteristics for the vehicle are proposed. Using the proposed characteristics, simulations were performed with the more advanced and proven DADS programme. The results show that the damper suggested by the optimisation study is indeed likely to improve the suspension of the vehicle. This study proves that the Vehsim2d/LFOPC vehicle modelling and optimisation system is indeed a valuable tool for a vehicle design team.     

Study on ride comfort of tractor with tandem suspension based on multi-body system dynamics

The aim of the paper is to report a systematic methodology which is used to evaluate and improve the ride comfort. An accurate model is necessary for further investigation and optimization. The vehicle dynamics model of tractor with tandem suspension is modeled and simulated in dynamics software ADAMS, which is redeveloped to add a function of automatic parametric modeling and simulation. The modeling methods of nonlinear characteristic components and various road excitation inputs, which can be simply seen as the implementation means for the model solution, are introduced. A new index called annoyance rate is presented to indicate the quantitative correlation between objective method and subjective comment. The quantitative correlation between them, which is quite different from the qualitative “comfortable” or “uncomfortable” results attained by objective evaluation, can be defined by function and regarded as a basis to scientifically evaluate and improve the ride comfort. According to the request of performance-based design, the parameter sensitive analysis and structure optimization have been carried out to find the trade-off among ride comfort, maneuverability and safety. The approach has proved to be very effective for predicting and improving the ride comfort by experiment results. The methodology can be also used for any other specific category of vehicle.     

Optimisation of road vehicle passive suspension systems. Part 1. Optimisation algorithm and vehicle model

Numerous problems have in the past been experienced during the development of military vehicle suspension systems. In order to solve some of these problems a two-dimensional multi-body vehicle dynamics simulation model has been developed for computer implementation. This model is linked to a mathematical optimisation algorithm in order to enable the optimisation of vehicle design parameters through the minimisation of a well defined objective function. In part 1 of this paper the concept of multi-disciplinary design optimisation is discussed. This is followed by the presentation of the up to six degrees of freedom vehicle model developed for this study, and a discussion of the specific gradient-based optimisation algorithm selected for the optimisation. In particular the derivation of the set of second-order differential equations, describing the acceleration of the different solid bodies of the vehicle model, is presented. In order to perform the optimisation of the non-linear suspension component characteristics, a six piece-wise continuous and linear approximation is used which is also described in this paper. Part 2 of this study will outline the simulation programme and the qualification of the programme. It will also present a typical case study where the proposed optimisation methodology is applied to improve the damper characteristics of a specific vehicle.     

Optimisation of maintenance scheduling strategies on the grid

The emerging paradigm of Grid Computing provides a powerful platform for the optimisation of complex computer models, such as those used to simulate real-world logistics and supply chain operations. This paper introduces a Grid-based optimisation framework that provides a powerful tool for the optimisation of such computationally intensive objective functions. This framework is then used in the optimisation of maintenance scheduling strategies for fleets of aero-engines, a computationally intensive problem with a high-degree of stochastic noise, achieving substantial improvements in the execution time of the algorithm.     

Vehicle dynamics simulations with coupled multibody and finite element models

With the advent of multibody system simulations (MSS) programs, it has become common practice to use computer modeling to evaluate vehicle dynamics performance. This approach has proved to be very effective for predicting the handling performance of vehicles; however, it has proved less successful for predicting the vehicle response at frequencies that are of interest in ride harshness and durability applications. The lack of correlation between theory and experiment can be partially traced back to tire models that are inadequate for rough road simulation. This paper presents a comprehensive vehicle dynamics model for simulating the dynamic response of ground vehicles on rough surfaces. This approach uses a MSS program to simulate the vehicle and a nonlinear FE program for the tires. Parallel processing of the tire models improves the efficiency of the overall simulation. Applications for this technology include vehicle ride and harshness analysis and durability loads simulation. This paper describes the MSS vehicle model, the tire FE model, and the interface which transfers data between the two simulations. Simulation and experiment results for a single tire without a vehicle encountering an obstacle and for a vehicle with four tires driving across a pot hole are presented. Conclusions and opportunities for further research end the paper.     

Simplified modelling of joints and beam-like structures for BIW optimization in a concept phase of the vehicle design process

The paper proposes an engineering approach for the replacement of beam-like structures and joints in a vehicle model. The final goal is to provide the designer with an effective methodology for creating a concept model of such automotive components, so that an NVH optimization of the body in white (BIW) can be performed at the earliest phases of the vehicle design process. The proposed replacement methodology is based on the reduced beam and joint modelling approach, which involves a geometric analysis of beam-member cross-sections and a static analysis of joints. The first analysis aims at identifying the beam center nodes and computing the equivalent beam properties. The second analysis produces a simplified model of a joint that connects three or more beam-members through a static reduction of the detailed joint FE model.In order to validate the proposed approach, an industrial case-study is presented, where beams and joints of the upper region of a vehicle's BIW are replaced by simplified models. Two static load-cases are defined to compare the original and the simplified model by evaluating the stiffness of the full vehicle under torsion and bending in accordance with the standards used by automotive original equipment manufacturer (OEM) companies. A dynamic comparison between the two models, based on global frequencies and modal shapes of the full vehicle, is presented as well.     

A meta-heuristic approach for solving the Urban Network Design Problem

This paper proposes an optimisation model and a meta-heuristic algorithm for solving the urban network design problem. The problem consists in optimising the layout of an urban road network by designing directions of existing roads and signal settings at intersections. A non-linear constrained optimisation model for solving this problem is formulated, adopting a bi-level approach in order to reduce the complexity of solution methods and the computation times. A Scatter Search algorithm based on a random descent method is proposed and tested on a real dimension network. Initial results show that the proposed approach allows local optimal solutions to be obtained in reasonable computation times.     

Hybrid evolutionary algorithm with Hermite radial basis function interpolants for computationally expensive adjoint solvers

In this paper, we present an evolutionary algorithm hybridized with a gradient-based optimization technique in the spirit of Lamarckian learning for efficient design optimization. In order to expedite gradient search, we employ local surrogate models that approximate the outputs of a computationally expensive Euler solver. Our focus is on the case when an adjoint Euler solver is available for efficiently computing the sensitivities of the outputs with respect to the design variables. We propose the idea of using Hermite interpolation to construct gradient-enhanced radial basis function networks that incorporate sensitivity data provided by the adjoint Euler solver. Further, we conduct local search using a trust-region framework that interleaves gradient-enhanced surrogate models with the computationally expensive adjoint Euler solver. This ensures that the present hybrid evolutionary algorithm inherits the convergence properties of the classical trust-region approach. We present numerical results for airfoil aerodynamic design optimization problems to show that the proposed algorithm converges to good designs on a limited computational budget.     

On the influence of design parameters and nonlinearities in vehicle suspensions on road deformation

Among others, two main objectives of modern vehicle design are road friendliness and ride comfort. Both aspects are strongly related since the dynamical tire forces depend on the vertical acceleration of the vehicle. In order to investigate the influence of design and operation parameters, different car models are considered which move with constant velocity on a rippled road. First, a linear half car model is examined and the influence of different design parameters is discussed. Second, nonlinear suspensions with Coulomb friction due to sealings as well as with bilinear shock absorbers are taken into account. The vertical dynamics of the vehicle model and the dynamic tire forces between vehicle and road are calculated using analytical methods. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fuzzy multi-objective optimization of passive suspension parameters

The paper proposed a systematic and effective optimization process to optimize a 3-D vehicle suspension dynamic model with eight DOF, including seat vertical motion, suspension vertical, pitching and rolling motions and wheels vertical motions using fuzzy optimization, to attain the best compromise between ride comfort and vehicle-generated road damage. The results show a substantial improvement in the vertical ride quality is obtained while keeping the suspension deflections within their allowable clearance when the vehicle moves at a constant velocity v = 20m/s, and the comfort performance of a suspension seat can be considerably enhanced.     

Subspace clustering for the finite mixture of generalized hyperbolic distributions

The finite mixture of generalized hyperbolic distributions is a flexible model for clustering, but its large number of parameters for estimation, especially in high dimensions, can make it computationally expensive to work with. In light of this issue, we provide an extension of the subspace clustering technique developed for finite Gaussian mixtures to that of generalized hyperbolic distribution. The methodology will be demonstrated with numerical experiments.

     

A generating set search method using curvature information

Direct search methods have been an area of active research in recent years. On many real-world problems involving computationally expensive and often noisy functions, they are one of the few applicable alternatives. However, although these methods are usually easy to implement, robust and provably convergent in many cases, they suffer from a slow rate of convergence. Usually these methods do not take the local topography of the objective function into account. We present a new algorithm for unconstrained optimisation which is a modification to a basic generating set search method. The new algorithm tries to adapt its search directions to the local topography by accumulating curvature information about the objective function as the search progresses. The curvature information is accumulated over a region thus smoothing out noise and minor discontinuities. We present some theory regarding its properties, as well as numerical results. Preliminary numerical testing shows that the new algorithm outperforms the basic method most of the time, sometimes by significant relative margins, on noisy as well as smooth problems. This work was supported by the Norwegian Research Council (NFR).     

An approximate algorithm for prognostic modelling using condition monitoring information

Established condition based maintenance modelling techniques can be computationally expensive. In this paper we propose an approximate methodology using extended Kalman-filtering and condition monitoring information to recursively establish a conditional probability density function for the residual life of a component. The conditional density is then used in the construction of a maintenance/replacement decision model. The advantages of the methodology, when compared with alternative approaches, are the direct use of the often multi-dimensional condition monitoring data and the on-line automation opportunity provided by the computational efficiency of the model that potentially enables the simultaneous condition monitoring and associated inference for a large number of components and monitored variables. The methodology is applied to a vibration monitoring scenario and compared with alternative models using the case data.     

Fast reconstruction of aerodynamic shapes using evolutionary algorithms and virtual nash strategies in a CFD design environment

This paper compares the performances of two different optimisation techniques for solving inverse problems; the first one deals with the Hierarchical Asynchronous Parallel Evolutionary Algorithms software (HAPEA) and the second is implemented with a game strategy named Nash-EA. The HAPEA software is based on a hierarchical topology and asynchronous parallel computation. The Nash-EA methodology is introduced as a distributed virtual game and consists of splitting the wing design variables-aerofoil sections-supervised by players optimising their own strategy. The HAPEA and Nash-EA software methodologies are applied to a single objective aerodynamic ONERA M6 wing reconstruction. Numerical results from the two approaches are compared in terms of the quality of model and computational expense and demonstrate the superiority of the distributed Nash-EA methodology in a parallel environment for a similar design quality.     

Using inexact gradients in a multilevel optimization algorithm

Many optimization algorithms require gradients of the model functions, but computing accurate gradients can be computationally expensive. We study the implications of using inexact gradients in the context of the multilevel optimization algorithm MG/Opt. MG/Opt recursively uses (typically cheaper) coarse models to obtain search directions for finer-level models. However, MG/Opt requires the gradient on the fine level to define the recursion. Our primary focus here is the impact of the gradient errors on the multilevel recursion. We analyze, partly through model problems, how MG/Opt is affected under various assumptions about the source of the error in the gradients, and demonstrate that in many cases the effect of the errors is benign. Computational experiments are included.     

Influence of Internal Actuator Properties of Active Anti-Roll Systems on the Vehicle Driving Behaviour

The application process of active anti-roll systems can be supported by the implementation of numerical optimization routines. The optimization objectives evaluate the handling and comfort vehicle behaviour and are calculated online from vehicle measurement variables. A model-based analysis illustrates the influence of the actuator properties on these optimization objectives. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Jump Phenomena and Bifurcations in Stochastic Vehicle-Road Dynamics

When vehicles ride on uneven roads, they are excited to vertical random vibrations whose stationary rms-values (root-mean-square) strongly depend on the velocity of the vehicle. To investigate this vibration behavior, it is appropriate to introduce road models in way domain which are based on the theory of stochastic differential equations and transformed from way to time by means of velocity-dependent way and noise increments. The random base excitations by roads are applied to nonlinear quarter car models. They lead to stationary rms-values of the vertical vehicle vibrations which become resonant for critical velocities and show jump phenomena similar to those of the Duffing oscillator under harmonic excitations. In the stochastic case, jump phenomena are only observable for narrow-banded road excitations. They vanish for increasing car damping and excitation bandwidth. For efficient simulations of the road-vehicle model, the n state equations are utilized to derive n(n + 1)/2 stochastic covariance equations. For small step sizes, their numerical mean square solutions coincide with the nonlinear results of fix-point iterations obtained when the noise terms of the covariance equations are omitted. It can easily be shown, that this deterministic approach leads to the correct stationary covariances in the linear case. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparative evaluation of some interactive reference point-based methods for multi-objective optimisation

Many real-world optimisation applications include several conflicting objectives of possibly nondifferentiable character. However, the lack of computationally efficient, interactive methods for nondifferentiable multi-objective optimisation problems is apparent. To satisfy this demand, a method called NIMBUS has been developed. Two versions of the basic method are presented and compared both theoretically and computationally. In order to give variety to the comparison, a related approach, called reference direction method is included. Theoretically, the methods differ in handling the information requested from the user. Numerical experiments indicate differences in computational efficiency and controllability of the solution processes.     

GOSAC: global optimization with surrogate approximation of constraints

We introduce GOSAC, a global optimization algorithm for problems with computationally expensive black-box constraints and computationally cheap objective functions. The variables may be continuous, integer, or mixed-integer. GOSAC uses a two-phase optimization approach. The first phase aims at finding a feasible point by solving a multi-objective optimization problem in which the constraints are minimized simultaneously. The second phase aims at improving the feasible solution. In both phases, we use cubic radial basis function surrogate models to approximate the computationally expensive constraints. We iteratively select sample points by minimizing the computationally cheap objective function subject to the constraint function approximations. We assess GOSAC’s efficiency on computationally cheap test problems with integer, mixed-integer, and continuous variables and two environmental applications. We compare GOSAC to NOMAD and a genetic algorithm (GA). The results of the numerical experiments show that for a given budget of allowed expensive constraint evaluations, GOSAC finds better feasible solutions more efficiently than NOMAD and GA for most benchmark problems and both applications. GOSAC finds feasible solutions with a higher probability than NOMAD and GOSAC.