Variational Methods in Sensitivity Analysis and Optimization for Aerodynamic Applications

The one- and two-dimensional inviscid Euler equations are formulated in an integro-differential form for shape design sensitivity analysis and optimization. The principal tool employed to derive the performance derivative sensitivity equations is the variational method, which is a continuous alternative to the discrete sensitivity analysis. Along with the sensitivity equations, the co-state equations and their boundary conditions are derived. Thereafter, based on the modal analysis of Von Neumann theory, the stability limits of the co-state equations are investigated. The stability characteristics of the co-state equations are compared with those of the Euler equations. Finally, using the criteria from the stability analysis of the state and co-state equations, some results of shape design optimization and sensitivity analysis for both quasi one- and two-dimensional Euler equations are presented. Copyright © 1999 John Wiley & Sons, Ltd.

PurposeThe objectives of this paper are the application of sensitivity analysis (SA) methods in atmospheric dispersion modeling to the emission dispersion model (EDM) to study the prediction of atmospheric dispersion of NO2 generated by an industrial fire, whose results are useful for fire safety applications. The EDM is used to predict the level concentration of nitrogen dioxide (NO2) emitted by an industrial fire in a plant located in an industrial region site in Algeria.Design/methodology/approachThe SA was defined for the following input parameters: wind speed, NO2 emission rate and viscosity and diffusivity coefficients by simulating the air quality impacts of fire on an industrial area. Two SA methods are used: a local SA by using a one at a time technique and a global SA, for which correlation analysis was conducted on the EDM using the standardized regression coefficient.FindingsThe study demonstrates that, under ordinary weather conditions and for the fields near to the fire, the NO2 initial concentration has the most influence on the predicted NO2 levels than any other model input. Whereas, for the far field, the initial concentration and the wind speed have the most impact on the NO2 concentration estimation.Originality/valueThe study shows that an effective decision-making process should not be only based on the mean values, but it should, in particular, consider the upper bound plume concentration.

Nearly all the computational time required in digital calculation of the gradient of an error index with respect to initial state is taken up by integration of differential equations. By direct application of the variational calculus, it is shown that the gradient may be calculated by integration of the state equations and costate equations, namely 2n equations. Previously, it has been thought necessary to integrate the n state equations and n2 state-sensitivity equations, namely n2+n equations.

A new analytical potential energy function for diatomic molecular ion XY+ is proposed based on the energy consistent method (ECM). The Coulomb potential included in the new ionic potential contains multipole corrections, converges quickly and is variationally, changeable. The new potential and the ECM are applied to variationally studying the potential energies of eight electronic states of several diatomic molecular ions: the A2π state of CO+, the X2∑g+ state of Li2+, the X2∑g+ state of He2+, the 12∏u state of Na2+, the A2∏u state of N2+, the X1∑+ state of KrH+, the X2∑+ state of SiO+ and the A2π state of SO+ ion. The present results agree excellently with the experiment-based Rydberg-Klein-Rees (RKR) potentials, and are superior to the commonly used Huxley-Murrell-Sorbie (HMS) analytical potentials, and are better in some cases than some quantum mechanicalab initio potentials in the ionic asymptotic and dissociation regions.

Design sensitivity analysis of structures deals with the calculation of the response derivatives with respect to the design variables. These derivatives, called the sensitivity coefficients, are used in the solution of various problems. In design optimization, the sensitivity coefficients are often required to select a search direction. These coefficients are used also in generating approximations for the response of a modified system. In addition, the sensitivities are required for assessing the effects of uncertainties in the structural properties on the system response. Calculation of the sensitivities involves much computational effort, particularly in large structural systems with many design variables. As a result, there has been much interest in efficient procedures for calculating the sensitivity coefficients. Developments in methods for sensitivity analysis are discussed in many studies (e.g. Haug et. al. 1986; Haftka and Adelman, 1989; Haftka and Gurdal, 1993; van Keulen et. al. in press). Methods of sensitivity analysis for discretized systems can be divided into the following classes: a. Finite-difference methods, which are easy to implement but might involve numerous repeated analyses and high computational cost, particularly in problems with many design or response variables. In addition, finite-difference approximations might have accuracy problems. The efficiency can be improved by using fast reanalysis techniques. b. Analytical methods, which provide exact solutions but might not be easy to implement in some problems such as shape optimization. c. "Semi-analytical" methods, which are based on a compromise between finite-difference methods and analytical methods. These methods use finite-difference evaluation of the right-hand-side vector. They are easy to implement but might provide inaccurate results. In general, the following factors are considered in choosing a suitable sensitivity analysis method for a specific application: the accuracy of the calculations, the computational effort involved and the ease-of-implementation. The implementation effort is weighted against the performance of the algorithms as reflected in their computational efficiency and accuracy. The quality of the results and efficiency of the calculations are usually two conflicting factors. That is, higher accuracy is often achieved at the expense of more computational effort. Dynamic sensitivity analysis has been demonstrated by several authors. Using the mode superposition approach and assuming harmonic loading, the response sensitivities were evaluated by direct differentiation of the equations of motion in the generalized coordinates (Kramer and Grierson, 1989). In cases of earthquake loading the ground acceleration is usually given in discrete time steps, thus the loading is not given analytically. In several studies (Kim and Choi, 2000; van Keulen et. al, in press) the unconditionally stable implicit numerical equation was directly derived. It was found that the analysis equations and the sensitivity equations have the same left-hand side expression. Thus, it was possible to use the available factorized coefficient matrix. A numerical procedure was applied for calculation of the sensitivity of the response.

A review of sensitivity analysis methods in building energy analysis

To improve the current industry standard one-parameter-at-a-time sensitivity analysis method, we propose a new sensitivity analysis framework that utilizes Plackett-Burman design and Random Forests (a well-known data mining method). The new framework significantly reduces the number of required simulation runs (i.e. samples), and at the same time compellingly improves the automatic history matching error. The proposed sensitivity analysis framework starts with generating samples/simulations using Plackett-Burman design. Here, each simulation is executed based on different combinations of the parameters' input values. Once the samples are ready, the parameters' input values and the target vector (i.e. the history matching error vector) are used to construct a Random Forests model. This model is used to rank the importance of each history matching parameter. Hence, parameters with low impact on the history matching error are discarded, and the remaining are used for Genetic Algorithm-based automatic history matching. The impact of an internal Random Forests parameter (number of decision trees) on the history matching error is also observed. The highly faulted reservoir with water injection is used for history matching and 10 parameters are defined uncertain consisting of fault transmissibilities, permeabilities, and connate water saturation. The aim is to match the bottom hole pressure and watercut for several oil producing wells. The one-parameter-at-a-time method requires 21 samples, and the selected top 4 parameters from this method are mainly fault transmissibilities. The resulting automatic history matching results gave a final error of 463.338. With the new framework, which requires only 12 samples instead of 21, the final error was 628.041, given that 100 trees are utilized. As we increased the number of trees to 500, the final error is significantly better (i.e. 36% improvement) than the results obtained from the one-parameter-at-a-time method. We can see that using the new framework, the majority of the top 4 parameters are related to permeabilities. Here, there is no significant change in terms of computation time as we increase the number of trees from 100 to 500; total computation time is less than 10 seconds. For this case study, not only this new framework requires significantly less number of initial simulation samples, it significantly improved the history matching error (as compared to the industry standard one-parameter-at-time sensitivity analysis method). Therefore, practicing engineers can utilize this framework to save time and simultaneously improve accuracy of the history matching parameter ranking. Parameter ranking computation with Random Forests takes less than 10 seconds.

It would be of interest to engineers and scientists to know the shape of the body of a given volume that will have minimum drag when moving through a viscous fluid at constant speed. It would be extremely useful if one could devise an evolution procedure that can evolve the minimum drag body in a logical and an orderly manner. Such a procedure was suggested by Pironneau for laminar flow wherein optimality conditions derived using optimal control theory were used in a non-linear gradient algorithm. The literature cites an attempt of the procedure at high Reynolds number where for each iteration in the evolution process, the flow field required an outer and an inner solution and the calculation of the gradient optimality condition required the solution of the co-state equation, a type of boundary layer equation. This paper addresses the direct simulation of the governing elliptic partial differential equations, viz., the Navier-Stokes and the co-state equations. Even though the latter has no simple mechanical interpretation, capitalizing on its resemblance to the former, this paper shows how the solution to the co-state equation could be obtained by simply adapting an existing Navier-Stokes code. Solution of the flow field and the calculation of the necessary criteria required in the evolution process are also discussed. The novelty of this direct approach is to make the evolution process more general, arbitrary and less complex. The profile evolution is demonstrated for flows at different Reynolds numbers.

Parturient paresis(milk fever) is a common disease associated with the onset of parturition in dairy cows. The disease is considered due to a large increased demand for calcium. Several work has mathematically and biologically modelled this process. Based on the existing models on calcium dynamics in diary cows, an optimal treatment is proposed in this paper. The treatment is executed at a time interval and lasts a relatively very small time duration, which is termed as a fixed time impulse control. For the optimization, with a selected objective function, a series of equations for optimality are to be satisfied, including control equations, costate equations and state equations. Those impulsive differential equations form a two point boundary value problem and are difficult to solve. A numerical scheme, SNAC(Single Network Adaptive Critic), is then proposed. The algorithm key is to use one neural network to capture the optimal relation between the pre-impulse state and the after-impluse costate. After the neural network is trained and the relation is captured, the optimal dosage of medicine can be provided when a parturient paresis is detected, and the cow's calcium level can be brought back to the normal status. Simulations are presented for illustrative purposes.

The uncertainty of parameters will have a significant impact on slope stability, where sensitivity analysis is a commonly used method in uncertainty research. However, traditional sensitivity analysis method costs much computation time. When calculating the sensitivity index of one parameter, all other parameters are taken as fixed values, and the uncertainty of all parameters cannot be considered simultaneously. Therefore, the variance-based and the moment-independent global sensitivity analysis (GSA) methods are both introduced to determine the influence of geotechnical parameters on slope stability in this study. To solve the importance index of GSA, the least angle regression algorithm, the kernel density estimation, and orthogonal polynomial estimation methods are developed to obtain variance-based importance index and the moment-independent importance index, respectively. The proposed methods allow all variables to change simultaneously within their variation range and have high computational efficiency. The results are in good at with those obtained by the variance-based Monte Carlo simulation method, which is considered as the exact solution forobtaining the importance index. The influence of the correlation between the shear strength parameters (c and φ) on the importance index is also studied, which indicates that the negative correlation will have a great impact on the importance index, which in turn affects the safety assessment of slope. Three engineering cases have been studied for engineering application, and the compared results indicate that the impact of the geotechnical parameters uncertainty on the safety factor (Fs) and failure probability (pf) are different. Therefore, the approaches based on GSA which can integrate the Fs with pf will be a promising approach for slope stability evaluation.

This paper is concerned with the shape sensitivity analysis of a class of boundary control, constrained, optimal control problems for parabolic systems. The notion of Euler and Lagrange derivatives of a boundary optimal control in the direction of a vector field is introduced. The derivatives are obtained in the form of optimal solutions of auxiliary optimal control problems. The method of sensitivity analysis used in this paper is based on related results on the differential stability of metric projections in Hilbert space onto a convex, closed subset, combined with the material derivative method of shape sensitivity analysis. Parabolic initial-boundary value problems with Dirichlet and Neumann boundary conditions are considered in this paper.

The problem of designing low-energy transfers between the Earth and the Moon has attracted recently a major interest from the scientific community. In this paper, an indirect optimal control approach is used to determine minimum-fuel low-thrust transfers between a low Earth orbit and a Lunar orbit in the Sun–Earth–Moon Bicircular Restricted Four-Body Problem. First, the optimal control problem is formulated and its necessary optimality conditions are derived from Pontryagin’s Maximum Principle. Then, two different solution methods are proposed to overcome the numerical difficulties arising from the huge sensitivity of the problem’s state and costate equations. The first one consists in the use of continuation techniques. The second one is based on a massive exploration of the set of unknown variables appearing in the optimality conditions. The dimension of the search space is reduced by considering adapted variables leading to a reduction of the computational time. The trajectories found are classified in several families according to their shape, transfer duration and fuel expenditure. Finally, an analysis based on the dynamical structure provided by the invariant manifolds of the two underlying Circular Restricted Three-Body Problems, Earth–Moon and Sun–Earth is presented leading to a physical interpretation of the different families of trajectories.

We study finite element approximations of parabolic optimal control problem with measure data in space in a bounded convex domain. The main mathematical difficulty of this kind of problem is low regularity of the solution of the state equation due to the presence of measure data in source term. This introduces some difficulties in both theory and numerics of finite element error analysis. We first prove the existence, uniqueness and regularity of the solution to the control problem. A priori error estimates for the state and control variables are derived for both the spatially discrete and fully discrete approximations of optimal control problems. Moreover, convergence properties for the state and the control variables are established. We use piecewise linear and continuous finite elements for the approximations of the state and co-state variables whereas the control variable is approximated by piecewise constant functions. Numerical experiment is provided to illustrate our theoretical results.

Optimizing building designs for energy efficiency and occupant comfort presents significant challenges due to the complex and often conflicting requirements of various stakeholders. Consequently, this study conducts a multifaceted sensitivity and economic impact analysis that aims to improve building performance in terms of energy efficiency and occupant comfort by implementing machine learning techniques. Using a broad dataset comprising of 12,000 energy simulation runs for Tvedestrand Upper Secondary School in Norway, several machine learning models were employed with Multi-Layer Perceptron outperforming the others. In addition, several sensitivity analysis methods were used to explore the influence of individual parameters on building performance. The analysis reveals that ventilation rate, room depth, U-value of the facade, and heat gains significantly affect energy consumption. Economic impact analysis was also carried out to compare the cost-effectiveness of traditional HVAC systems with Building Management System (BMS) HVAC solutions. The BMS HVAC system shows significantly lower operational costs over time, with investment costs averaging around 1200 Norwegian kroner (NOK)/m2 and operational costs of approximately 150 NOK/m2 per year. Sensitivity analysis under different economic scenarios highlights the economic viability of the BMS HVAC system. This study identifies optimal building parameters that balance energy efficiency and thermal comfort, achieving total energy consumption between 11.05 and 22.51 kWh/m2 and zero discomfort hours (h > 26 °C). In sum, the findings offer valuable insights for stakeholders, enabling informed decisions about sustainable building design and energy efficiency improvements, ensuring both technical soundness and financial viability under a wide range of conditions, while using the tested tools.

The large number of input factors involved in a sophisticated geotechnical computational model is a challenge in the concept of probabilistic analysis. In the context of model calibration and validation, conducting a sensitivity analysis is substantial as a first step. Sensitivity analysis techniques can determine the key factors which govern the system responses. In this paper, three commonly used sensitivity analysis methods are implemented on a sophisticated geotechnical problem. The computational model of a compressed air energy storage, mined in a rock salt formation, includes many input parameters, each with large amount of uncertainties. Sensitivity measures of different variables involved in the mechanical response of the cavern are computed by different global sensitivity methods, namely, Sobol/Saltelli, Random Balance Design, and Elementary Effect method. Since performing sensitivity analysis requires a large number of model evaluations, the concept of surrogate modelling is utilised to decrease the computational burden. In the following, the accuracy levels of various surrogate techniques are compared. In addition, a comparative study on the applied sensitivity analysis methods shows that the applied sensitivity analysis techniques provide identical parameter importance rankings, although some may also give more information about the system behaviour.