Conventional Response Surface Method Research Articles

Uncertainty analysis of mechanical dynamics by combining response surface method with signal decomposition technique

Studying multibody dynamic systems, a common way to evaluate the effects of uncertainty parameters is the response surface method, which works by building a polynomial surrogate model at each time instant through a set of input and corresponding output responses. However, a dynamic response always becomes a more and more complicated function of uncertainty parameters with the increase of time; as a result, the fitted surrogate model often fails to provide an accurate approximation at later time instants. This paper suggests that a dynamic response is composed of several vibration components and one trend component. Therefore, it is better to decompose the response into its multiple components, and then fit the amplitude and phase of each vibration component, as well as the trend component, using polynomial functions. By combining the response surface method with signal decomposition techniques, such as Hilbert-Huang transform and local mean decomposition, this study proposes a novel methodology to develop a high-accuracy surrogate model for interval uncertainty analysis. The proposed methodology will degenerate to the conventional response surface method when the dynamic response is a trend component. Three mechanical systems – a Duffing oscillator, a single pendulum and a slider-crank model – are used to confirm the effectiveness of the proposed methodology. This methodology provides a new technical pathway to improve the accuracy of dynamical uncertainty analysis.

Hybrid Approach for Optimizing Process Parameters in Biodiesel Production from Palm Oil

Biodiesel was synthesized from direct transesterification of palm oil reacted with methanol in the presence of a suitable catalyst. There is a sequence of three consecutive reversible reactions for the transesterification process. These process parameters were optimized via the hybrid optimization approach of a conventional response surface method and artificial intelligence mechanisms from Sine Cosine and Thermal Exchange Optimization metaheuristics. The influential parameters and their combined interaction effects on the transesterification efficiency were established through a factorial designed experiments. In this study, the influential parameters being optimized to obtain the maximum yield of biodiesel were reaction temperature of 60–150°C, reaction time of 1–6 hours, methanol to oil molar ratio of 6:1–12:1 mol/mol and weight of catalyst of 1–10wt. %. On the first phase, the analysis of variance (ANOVA) revealed the reaction time as the most influential parameter on biodiesel production. Based on the experimental results from the hybrid algorithm via the SCO, it was concluded that the optimal biodiesel yield for the transesterification of palm oil were found to be 100°C for reaction temperature, 4 hours for reaction time, 10:1 wt/wt of ratio methanol to oil and 8% of weight of catalyst with 92.15% and 90.97% of biodiesel yield for expected and experimental values, respectively.

Multi-objective robust design of vehicle structure based on multi-objective particle swarm optimization

With the continuous spread of COVID-19 epidemic, the strict control of personnel makes it a problem to optimize the design of vehicle parameters after field measurement. The energy absorption characteristics and deformation mode of the front structure of the vehicle determine the acceleration or force response of the vehicle body during the impact, which plays an important role in occupant protection. The traditional multi-objective optimization method is to transform multi-objective problems into single objective optimization problems through weighted combination, objective planning, efficiency coefficient and other methods. This method requires a strong prior knowledge. The purpose of this paper is to combine the experimental design with the Multi-objective Particle Swarm Optimization (MPSO) method to achieve the optimization of the crash worthiness of automobile structure. This method can effectively overcome the defect of low precision caused by the conventional response surface method in the whole design space. In this paper, the multi-objective particle swarm optimization method is applied to the research of Crash worthiness optimization of automobile structure, which expands the application field of the multi-objective particle swarm optimization method, and also has a very big role in the optimization of other complex systems. It can be seen from the experiment that the speed of multi-objective particle swarm optimization is much faster than that of other methods. Only 100 iterations can get the relative better results. The case study on the front structure of a car shows that the method has a good result. It is of great significance to apply the method to the optimization design of the crash worthiness of the car structure to improve the crash safety of the car under the influence of COVID-19 epidemic.

Surface work-hardening optimization of cold roll-beating splines based on an improved double-response surface-satisfaction function method

The work hardening of a spline during cold roll-beating is used as an indicator to evaluate the mechanical properties of the surface. To further optimize the work-hardening degree of a cold roll-beating spline surface, weight theory and satisfaction functions are used to improve the double-response surface-satisfaction function model. The model describes the involute spline based on the cold roll-beating speed and feed rate. The generalized reduced-order gradient method is applied to optimize the optimal combination of processing parameters. The experiments validate the optimization results of the improved double-response surface-satisfaction function method and the conventional response surface method based on the cold roll-beating spline test and a comparative analysis of the spline surface metallographic structure. The results show that the satisfaction degree of the improved response model is 0.87384, indicating that the model is robust and reliable. The optimized processing parameters are a cold roll speed of 1448.21 r/mm, a feed rate of 41.71 mm/min, and a degree of work hardening of 144.79%. The spline surface work-hardening degree based on the revised model is higher than that of the conventional model. Thus, the improved double-response surface-satisfaction function model provides better accuracy.

Reliability analysis of a large-scale landslide using SOED-based RSM

A design matrix in response surface method (RSM) that satisfies the orthogonality is very useful because the mean square error can be minimized, so that the response surface is more precise. But the orthogonality of a second-order design matrix in conventional RSM cannot be satisfied. In this paper, a second-order orthogonal experimental design (SOED)-based RSM is proposed by considering the orthogonality of high-order design matrix. The SOED is constructed by changing the length of star points, and the main characteristic of SOED is that the design matrix is diagonal. When the high-order terms are considered in the SOED-based RSM, a globe optimal solution can be found. As the regression equation is determined, the reliability index can be analyzed by the normalized distance between the mean value of the performance function and the critical limit state of the safety factor. A practical large-scale landslide with two slip surfaces is taken to verify the applicability and precision of the proposed method in detail. It is found that the SOED-based response surface is more rigorous than the conventional RSM.

Robust design optimisation via surrogate network model and soft outer array design

Robust design searches for a performance optimum with least sensitivity to variable and parameter variations. Taguchi method applies an inner array for control factors and an outer array for noise factors to estimate the Signal-to-Noise ratio (S/N). However, the cross product arrays impose serious cost concerns for expensive samplings. Also, rigorous control of noise factors to pre-set levels is impractical in industrial applications. This study presents a soft computing-based robust optimisation that merges control and noise factors into a combined experimental design to establish a surrogate using artificial neural network. Genetic algorithm is applied to search in the sub-space of control factors in the surrogate with a soft outer array to estimate the S/N served as the evolution fitness. Performance variations due to the tolerances of control and uncontrollable factors can then be estimated without conducting actual experiments. The verifications of the predicted optima become additional learning samples to refine the surrogate, and the iteration continues until convergence. The robust optimisation of a micro-accelerometer with maximised gain is used as an illustrative example. The proposed algorithm provides a superior robust optimum using a much smaller sample and less controlling cost compared with Taguchi method and a conventional response surface method.

A hybrid approach combining uniform design and support vector machine to probabilistic tunnel stability assessment

Applications of the reliability-based method to stability evaluation of tunnel structures have become an ever-increasing concern over recent years. One critical challenge in conducting such a task is the implicit nature of the limit state function (LSF). To address this issue, the focus of this study is on, among others, the use of response surface method (RSM) by considering both the selection of the sampling method and the choice of the response surface form (as two major factors affecting the RSM’s performance). In this context, the current paper develops for tunnel-reliability analysis a hybrid approach combing an experimental design called uniform design (UD) and a regression device known as support vector machine (SVM). For the proposed hybrid approach, the UD is used to generate sampling points and then the SVM is employed to construct the response surface approximating the original inexplicit LSF. Such an approach integrates the merits of both UD and SVM used for complex nonlinear modelling. Three carefully selected tunnel examples are illustrated: one for a typical tunnel under relatively simplified tunnelling conditions and the other two for real-life tunnels. Comparisons are made to validate the computational accuracy and efficiency of the present approach. In particular, for the tunnel example where the LSF is known only implicitly through the numerical analyses (which is the scenario of many real-world applications in tunnel community), the obtained results further demonstrate the efficiency of this approach: it can be much more economical to achieve reasonable accuracy than the conventional RSMs when a small number of sampling data is used. Such comparisons made in this work verify the application potential of the developed hybrid approach for probabilistic tunnel stability assessment involving the implicit LSF.

Structural Reliability Analysis for Implicit Performance with Legendre Orthogonal Neural Network Method

In order to evaluate the failure probability of a complicated structure, the structural responses usually need to be estimated by some numerical analysis methods such as finite element method (FEM). The response surface method (RSM) can be used to reduce the computational effort required for reliability analysis when the performance functions are implicit. However, the conventional RSM is time-consuming or cumbersome if the number of random variables is large. This paper proposes a Legendre orthogonal neural network (LONN)-based RSM to estimate the structural reliability. In this method, the relationship between the random variables and structural responses is established by a LONN model. Then the LONN model is connected to a reliability analysis method, i.e. first-order reliability methods (FORM) to calculate the failure probability of the structure. Numerical examples show that the proposed approach is applicable to structural reliability analysis, as well as the structure with implicit performance functions.

Stochastic free vibration analysis of laminated composite plates using polynomial correlated function expansion

Over the past few decades, composite materials are extensively used in various industries due to their high specific stiffness, strength, weight sensitivity and cost-effectiveness. However, lack of complete control over the manufacturing process results in undesirable uncertainties, which in turn affect the vibrational characteristics of systems. This paper presents a novel approach, referred to as polynomial correlated function expansion (PCFE), for stochastic free vibration analysis of composite laminate. The proposed approach facilitates a systematic mapping between the input and output variables by expressing the output in a hierarchical order of component functions. The component functions are expressed in terms of extended bases and the unknown coefficients associated with the bases are determined by employing homotopy algorithm. The proposed approach has been employed for stochastic free vibration analysis of laminated composite plates. Results obtained using PCFE have been compared with results obtained using radial basis function (RBF) and conventional response surface method (RSM). Compared to RBF and RSM, PCFE yield more accurate result from considerably fewer sample points. Furthermore, case studies with different ply orientations have also been performed. Based on the numerical results, new physical insights have been drawn on dynamic behaviour of composite laminates.

반응표면법을 이용한 DTF의 석탄 연소 안전성 평가

The experimental design methodology was applied in the drop tube furnace (DTF) to predict the various combustion properties according to the operating conditions and to assess the coal plant safety. Response surface method (RSM) was introduced as a design of experiment, and the database for RSM was set with the numerical simulation of DTF. The dependent variables such as burnout ratios (BOR) of coal and <TEX>$CO/CO_2$</TEX> ratios were mathematically described as a function of three independent variables (coal particle size, carrier gas flow rate, wall temperature) being modeled by the use of the central composite design (CCD), and evaluated using a second-order polynomial multiple regression model. The prediction of BOR showed a high coefficient of determination (R2) value, thus ensuring a satisfactory adjustment of the second-order polynomial multiple regression model with the simulation data. However, <TEX>$CO/CO_2$</TEX> ratio had a big difference between calculated values and predicted values using conventional RSM, which might be mainly due to the dependent variable increses or decrease very steeply, and hence the second order polynomial cannot follow the rates. To relax the increasing rate of dependent variable, <TEX>$CO/CO_2$</TEX> ratio was taken as common logarithms and worked again with RSM. The application of logarithms in the transformation of dependent variables showed that the accuracy was highly enhanced and predicted the simulation data well.

Reliability Analysis of Automobile Engine Connecting Rod Using Fourier Orthogonal Neural Network Method

In order to predict the failure probability of a complicated structure, the structural responses usually need to be estimated by a numerical procedure, such as finite element method (FEM). To reduce the computational effort required for reliability analysis, response surface method could be used. However the conventional response surface method is still time consuming especially when the number of random variables is large. In this paper, a Fourier orthogonal neural network (FONN)-based response surface method is adopted to solve the reliability analysis of the automobile engine. The working process of the connecting rod is simulated with UG software, the dynamics analysis on crank-connecting rod-piston mechanism is performed with ANSYS and ADAMS software, with FEM analysis results, the stress information of the critical point of the structure can be obtained, so the performance function of the structure can be established. The FONN method is used to fit the performance function as well as its derivatives, so as to calculate the reliability of the structure.

Structural Reliability Analysis Using Legendre Orthogonal Neural Network Method

In order to predict the failure probability of a complicated structure, the structural responses usually need to be estimated by a numerical analysis such as finite element method. The response surface method could be used to reduce the computational effort required for reliability analysis when the performance functions are implicit. However the conventional response surface method is time-consuming or cumbersome if the number of random variables is large. This paper presents a Legendre orthogonal neural network (LONN)-based response surface method to predict the reliability of a structure. In this method, the relationship between the random variables and structural responses is established by a LONN model. Then the LONN model is connected to a reliability method, i.e. first-order reliability methods (FORM) to predict the failure probability of the structure. Numerical example has shown that the proposed approach is applicable to structural reliability analysis involving implicit performance functions.

Structural Reliability Analysis Using Fourier Orthogonal Neural Network Method

In order to predict the failure probability of a complicated structure, the structural responses usually need to be predicted by a numerical procedure, such as FEM method. The response surface method could be used to reduce the computational effort required for reliability analysis. However the conventional response surface method is still time consuming when the number of random variables is large. In this paper, a Fourier orthogonal neural network (FONN)-based response surface method is proposed. In this method, the relationship between the random variables and structural responses is established using FONN models. Then the FONN model is connected to the first order and second moment method (FORM) to predict the failure probability. Numerical example result shows that the proposed approach is efficient and accurate, and is applicable to structural reliability analysis.

Multivariate adaptive regression splines based simulation optimization using move-limit strategy

This paper makes an approach to the approximate optimum in structural design, which combines the global response surface (GRS) based multivariate adaptive regression splines (MARS) with Move-Limit strategy (MLS). MARS is an adaptive regression process, which fits in with the multidimensional problems. It adopts a modified recursive partitioning strategy to simplify high-dimensional problems into smaller highly accurate models. MLS for moving and resizing the search sub-regions is employed in the space of design variables. The quality of the approximation functions and the convergence history of the optimization process are reflected in MLS. The disadvantages of the conventional response surface method (RSM) have been avoided, specifically, highly nonlinear high-dimensional problems. The GRS/MARS with MLS is applied to a high-dimensional test function and an engineering problem to demonstrate its feasibility and convergence, and compared with quadratic response surface (QRS) models in terms of computational efficiency and accuracy.

Multi-variable thermal design of T-structured phase-change memory cell using advanced response surface method

Thermal design is crucial in designing phase-change memory (PCM) as it is operated by thermal energy. Among the numerous factors that influence the performance of PCM, geometrical configuration is the most critical factor as the heat confinement is determined by the effective thermal resistance of the PCM cell. This study reports the thermal design of T-structured PCM cell with contact diameter of 80nm. Five design variables, which are the thicknesses of two metal contact layers, two metal electrode layers, and one phase-change layer, are studied. Cell performance is evaluated in terms of the highest cell temperature during the reset process. The performance results of the five variables are correlated into a single equation using the advanced response surface method, which is a modified form of conventional response surface method. By doing this, one can easily design and define the PCM with the best performance under given constraints. Given design ranges, results show that the cell performance is maximum when the two top metal layers and the bottom metal electrode become thinner while the bottom metal contact becomes thicker. However, there exists a certain phase-change layer thickness, which is 36nm in this study, for the maximum performance to occur.

An efficient response surface method using moving least squares approximation for structural reliability analysis

The response surface method (RSM) is widely adopted for structural reliability analysis because of its numerical efficiency. However, the RSM is time consuming for large-scale applications and sometimes shows large errors in the calculation of the sensitivity of the reliability index with respect to random variables. In order to overcome these problems, this study proposes an efficient RSM applying a moving least squares (MLS) approximation instead of the traditional least squares approximation generally used in the RSM. The MLS approximation gives higher weight to the experimental points closer to the most probable failure point (MPFP), which allows the response surface function (RSF) to be closer to the limit state function at the MPFP. In the proposed method, a linear RSF is constructed at first and a quadratic RSF is formed using the axial experimental points selected from the reduced region where the MPFP is likely to exist. The RSF is updated successively by adding one new experimental point to the previous set of experimental points. Numerical examples are presented to demonstrate the improved accuracy and computational efficiency of the proposed method compared to the conventional RSM.

Support vector machine for structural reliability analysis

Support vector machine (SVM) was introduced to analyze the reliability of the implicit performance function, which is difficult to implement by the classical methods such as the first order reliability method (FORM) and the Monte Carlo simulation (MCS). As a classification method where the underlying structural risk minimization inference rule is employed, SVM possesses excellent learning capacity with a small amount of information and good capability of generalization over the complete data. Hence, two approaches, i.e., SVM-based FORM and SVM-based MCS, were presented for the structural reliability analysis of the implicit limit state function. Compared to the conventional response surface method (RSM) and the artificial neural network (ANN), which are widely used to replace the implicit state function for alleviating the computation cost, the more important advantages of SVM are that it can approximate the implicit function with higher precision and better generalization under the small amount of information and avoid the “curse of dimensionality”. The SVM-based reliability approaches can approximate the actual performance function over the complete sampling data with the decreased number of the implicit performance function analysis (usually finite element analysis), and the computational precision can satisfy the engineering requirement, which are demonstrated by illustrations.

Reliability analysis of structures using neural network method

In order to predict the failure probability of a complicated structure, the structural responses usually need to be estimated by a numerical procedure, such as finite element method. To reduce the computational effort required for reliability analysis, response surface method could be used. However the conventional response surface method is still time consuming especially when the number of random variables is large. In this paper, an artificial neural network (ANN)-based response surface method is proposed. In this method, the relation between the random variables (input) and structural responses is established using ANN models. ANN model is then connected to a reliability method, such as first order and second moment (FORM), or Monte Carlo simulation method (MCS), to predict the failure probability. The proposed method is applied to four examples to validate its accuracy and efficiency. The obtained results show that the ANN-based response surface method is more efficient and accurate than the conventional response surface method.

General Response Surface Reliability Analysis for Fuzzy-Random Uncertainty both in Basic Variables and in State Variables

A general response surface(RS) method is presented for reliability analysis of complex structure/mechanism with fuzzy-random uncertainty both in basic variables and in failure state variables. On the basis of equivalent transformation from fuzzy basic variable to random basic variable, the fuzziness and randomness in the basic variables are considered simultaneously in the presented general RS method. Once the fuzzy basic variables are transformed into the random basic variables, the conventional RS method is employed to establish the general RS for the complex structure/mechanism with implicit limit state equation by finite element numerical simulation. Furthermore, the general failure probability is defined according to the probability formula for fuzzy-random event by taking the fuzziness and randomness in the failure safety state into consideration, and an appropriate fuzzy operator is adopted to calculate the general failure probability for the complex structure/mechanism with multiple implicit failure modes. Finally, a general reliability analysis of an elastic linkage mechanism is introduced to illustrate the present method.