Eklenik değişken yöntemi ile malzeme, yük. biçim duyarlılık çözümlemesi ve en iyileme
Material, load and shape sensitivity analysis and optimization by the adjoınt variable method
- Tez No: 46513
- Danışmanlar: PROF.DR. R. ALSAN MERİÇ
- Tez Türü: Doktora
- Konular: Astronomi ve Uzay Bilimleri, Astronomy and Space Sciences
- Anahtar Kelimeler: Isı iletimi, Optimizasyon modelleri, Heat conduction, Optimization models
- Yıl: 1995
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Bu çalışmada, integral denklemlerle tanımlı doğa olaylarını içeren eniyileme problemlerine ait duyarlılık çözümlemeleri, eklenik değişken yöntemi ile gerçekleştirilmektedir. Bu işlem sırasında öncelikle diferansiyel denklemi bilinen bir doğa olayı (katı cisimlerde ısı iletimi) için diferansiyel duyarlılık çözümlemesi yapılmakta, ardından aynı olgunun sınır integral denklemi için integral duyarlılık çözümlemesi gerçekleştirilmektedir. Bu çözümlemeler, katı cismin malzemesi, yüklemesi ve biçimi (integral hariç) için yapılmaktadır. Çalışma süresince gerçekleştirilen uygulamalarla, önerilen duyarlılık çözümlemesinin güvenirliği iki ayrı geometride denenmektedir. Çözümlemeler, kesin çözümü bilinen bir problemde ve teknolojide pratik yararı olan bir geometride ayrı ayrı denenmektedir. Sonuçlar göstermektedir ki, bir diferansiyel denkleme ait duyarlılık çözümlemesi, hem bu denklemin kendisini kullanan diferansiyel yöntemle hem de aynı denklemin sınır integral denklemini kullanan integral yöntemle aynı doğrulukta sonuçlan oluşturmaktadır. Bu nedenle integral duyarlılık çözümlemesi, integral denklemlerle betimli sistemlerde güvenilerek kullanılabilir.
Özet (Çeviri)
SUMMARY MATERIAL, LOAD AND SHAPE SENSITIVITY ANALYSIS AND OPTIMIZATION BY THE ADJOINT VARIABLE METHOD In this study, material, load and shape sensitivity analysis is considered. Sensitivities for optimization problem of heat conducting solid bodies are obtained by the adjoint variable method. Related applications are discussed for differential and integral equation based systems. The main purpose of optimization techniques is to achive the best or the most effective solution. In optimization problems, it is necessary to define a functional which determines the behavior of the system. These functionals are called“objective functionals”and they define the reaction of the system during optimization. It is also important to define the technological and/or geometrical constraints in the design problems. The functionals which define these limits are called“constraint functionals”. To obtain a feasible solution, the constraints should not be violated. In order to improve the design in the optimization process, it might be necessary to change (adjust) some of the variables in the system which are called“decision variables”.They can be classified in two groups. The first types are constants of the system which are known as“decision parameters”. The second types are“decision functions”which are functions of coordinates and/or time. The quantity of the change in the functional due to the change in a decision variable gives the sensitivity of the functional. To obtain a convergence to an optimal design, the sensitivities have to be calculated. By using this information, the required changes in the design variables can be determined. In special cases it is possible to use a function as an objective functional. In this case, the sensitivity of the function to a design parameter can be determined by differentiating the objective function with respect to design parameter. There are four types of optimization problems in engineering: material, load,shape and eigenvalue optimizations. In the material optimization, the physical material properties such as Young's modulus and heat transfer coefficients are determined. Load optimization problems involve adjusting the loading of the system for the desired design conditions. In shape optimization, the shape of the system needs to be determined for the optimum design. Finding proper eigenvalues of the vibration and the stability of the system requires eigenvalue optimization. For an efficient optimization process, the designer needs to satisfy following conditions: - A proper analysis of the system: to understand the behaviour of the system/ - A proper optimization algorithm: to obtain a convergence to the optimal design point. - A proper choice of the decision variables: to control the system. vi- Correct sensitivity infirmation: to determine the orientation of the design vector to reach the optimal point in a reasonable number of iterations. The engineering problems are usually defined in the implicit forms. It is therefore usually quite hard to get an explicit form of the field variables of the system. The object and/or constraint functionals are also calculated by integrating some functions in which the arguments are explicitly defined field variables: = / f(u, u,i )dü + g(u,u,i)dr Jn Jv where 0, Y and u define the domain of the system, boundary of the system' and the field variable respectively. In this equation, f and g are known functions of field variables. As the field variables are defined in implicit form, the sensitivities of the functionals can not be determined explicitly. Hence, discrete approximation techniques for the solution of the system and sensitivity analysis need to be used. There are three different approaches in sensitivty analysis: - Discretization and finite difference - Discretization and sensitivity analysis (direct differentiation) - Sensitivity analysis and discretization (adjoint variable method) In the finite differencing approach, the decision parameters need to be perturbed in the analysis step. Using the perturbed functional and forward differencing, sensitivities of a function (say G) can be determined: dG _G(Pi + APi)-G(Pi) dpi Api where pi + Api reperesents the perturbed decision parameter. This method has two main disadvantages: Once the number of parameters incereases, the number of analyses also increases. The sensitivity quantities are directly related with the magnitude of the perturbations in the parameters and the direction of the differencing. As a result of these disadvantages, the finite differencing approach is not a proper way for sensitivity analysis of multi-parameter systems. Another approach in sensitivity analysis is direct differentiation. In this method, the system is discretized and then the field variables are obtained in terms of the decision variables: [A]{fd} = {ud} where A is a matrix that explicit function of design parameters, fd is the discrete values of an effecting function, and uj is the discrete values of the field variable. The sensitivities are obtained by the chain rule. First the integrand which consists of / and g functions are differentiated with field variable. Then the field variable viiis differentiated with design parameter. The sensitivities are obtained by the integration of these differentiations. dy f“ E?=1 \dpAV x> The direct differentiation method has also some disadvantages. The field variables are obtained by an approximate method. Hence, the numerical error in the sensitivity expressions are directly related to the approximate solutions. The accuracy of the sensitivity expressions also depends on the method of the approximate solutions. The last approach for the Sensitivity Analysis is essentially using the so-called adjoint variable method. This method uses the governing equation of the physical system as a ”natural constraint“ equation. Thus, the system equation is incorporated into the relevant functionals by using an adjoint function as Lagrange multiplier function. During this operation there are two possibilities that can be applied. If partial differential equations are taken as the system (primary) equations, the method leads to the so-called ”Differential Formulation“ of Sensitivity Analysis. On the other hand, if integral equations are adopted as the system equations, then the method is called the ”Integral Formulation“ of Sensitivity Analysis. The integral equations may arise naturally in some physical phenomenon, such as heat radiation, or partial differential equations may be converted into (boundary) integral equations. During this study, the phenomenon to be analysed is the steady-state heat conduction in solids. The governing equation is in Poisson's equation form: ku,a +Q = 0 where, k is the heat conduction coefficient, u refers to the temperature, and Q represents the heat generation rate per unit volume within the solid. The problem may have three types of boundary conditions: - Prescribed temperature - Prescribed heat flux - Convection £er”teru u = u0 -kdu/dn = q0 -kdu/dn = h(u - u^) where u0 is the prescribed temperature, q0 indicates the heat flux, h is the heat coefficient and Moo refers to the ambient temperature. The boundary integral equation (BIE) form of this differential formulation is given by the following : c(0«U) + / 9* (f, x)u(x)dT(x) - J u'tf, x)q{x)dT{x) - I u*{i,x)%{x)d£l{x) =0 Jn k where, u* is the fundamental solution, q* is the normal derivative of the fundamental solution; T is the system boundary, and Ü is the system domain. viiiThe points of £ and x are observation point and integration point respectively. The explicit forms of the r,-, u*,q* and c(£) are given below: In certain cases it is possible to get rid of the domain integral involving the heat source by a transformation into a boundary integral. These special cases are as follows : Point sources, constant sources, harmonic sources and biharmonic sources. During this study, the last three cases are used for both the primary (governing) and adjoint equations. In particular, for a constant source a simple transformation of the dependent variable leads to the Laplace's equation without any source as the system equation. In case of harmonic and biharmonic sources, it is shown that the domain integral may be eliminated by using higher order of auxiliary fundamental solutions. The main aim of the thesis is the sensitivity analysis by several formulations. The sensitivities of the functionals, that are obtained by the adjoint variable method, are used by available optimization codes. At first, an unconstrained optimizer code is used for material and load optimizations. A constrained optimization routine is then used for shape optimization. These codes decide a new design point to converge to the optimal solution by a series of iterations. The new design point is determined as: x9 = xq~l + aSq where, q is the iteration number, S9 refers to the search vector, and a represents the scalar moving parameter. Sq and a quantity are computed by using the sensitivities. The material sensitivity analysis and load sensitivity analysis are quite identical. In both cases, the geometry is fixed and the relevant material property and/or loading functions appear directly in the governing equations or boundary conditions. On the other hand, the shape sensitivity analysis performed for a varying geometry requires the so-called generalized calculus of variations. This is done by using“the material derivative concept”which will be mentioned later. The differential material and load sensitivity analyses consist of six steps. 1. Augment the functional by incorporating the governing equation into the objective functional / in terms of an adjoint function ü. Î = 1 + / ü{ku,ü +Q)dü 2. Take the first variation of the augmented functional /. 6Î = 81+ / ü8(ku,u +Q) + (ku,u +Q)8üdÜ ix3. Integrate by parts some terms of the functional to get rid of high order derivatives. 4. Take the variations of the boundary conditions. £ e Tu : 8u = 8u0 £ <E Tu : Sdu/dn = -l/kSq0 £ G Tu : Sdu/dn = -h/kSu + h/kSuoo + 1/&(U°° - u)6h -h/tffaoo - u)6k 5. Define an adjoint problem by the undemanded variation terms of the primary variables 6u and 6um. 6. Write the sensitivity coefficients of the the decision variables with the remained terms of the variation of the functional. The integral sensitivity analysis procedure has similar procedure. 1. Augment the functional by incorporating boundary integral equation at the boundary (S) in terms of the û and boundary integral equation at the domain (D) in terms of another adjoint function Ü. 1 = 1 + f üDdÜ + + I ûSdT Ju Jr 2. Take the first variation of the augmented functional. 3. Apply the integration by parts to reduce the differentiations. 4. Interchange the integration order and the name of the dummy variables to take the variation terms from internal integrals to the external integrals. 5. Take the variation of the boundary conditions. 6. Define the adjoint problem (this problem will be defined by boundary integral equations) to get rid of the terms that undemanded. 7. Find and write the sensitivity coefficient expressions by the terms left at the variation of functional. The shape sensitivity analysis (SSA) uses the concept of material derivative. SSA is applied for the differential formulation. The procedure for the shape sensitivity analysis is given below: 1. Define the governing equation and the boundary conditions of the system. 2. Decide the objective criterion functional and decision variables of the system. 3. Augment the functional by incorporating the governing equation in terms of the adjoint function ü (same as in the material and load SA). 4. Apply the integration by parts to reduce the differentiations. 5. Take the material derivative of the variation of the functional. $ = / / dn =? $ = f fdn + [ fvndr * = J gdr-, => *=[</ + (9m +Hg)Vn]dr6. Apply the integration by parts again. 7. Take the material derivative of the boundary conditions 8. Define an adjoint problem. 9. Define the shape sensitivity expression The numerical efforts are made after the theoretical formulations. Applications are realized for two systems: - A geometry that the system solution and the sensitivities can be obtained analyticaly. It is an infinite cylinder with a hole inside (Figure I). The mathematical description of the system is given as: Figure 1. The cylinder with hole. - A turbine blade with two cooling holes inside (Figure II). For the cylindirical system, All numerical calculations are compared by the exact analytical solutions of the field variable u and u,n. The results of the computations have confirmed that the boundary element analysis code is robust for the systems described by the Poisson or Laplace equations. The adjoint functions at the boundaries for differential formulation are also compared with their exact solution. The comparisions have given quite satisfactory results for adjoint functions. On the other hand, some of the integral relations are obtained between the differential adjoint functions ü and, integral adjoint functions û, û. These relations are satisfied during the post processing phase. The integral relations between ü and û, ü are also obtained in the blade problem. Figure 2. The turbine blade with cooling holes. XIThe calculated functionals are compared by the exact solution. The results have reconfirmed the robustness of the boundary element code. Also, the quantities of the sensitivity coefficients obtained by both differential and integral sensitivity analysis are compared by three other type of sensitivities: The exact solution of sensitivities (if possible) The finite difference calculation for sensitivities A chosen parameter is increased by constant perturbation iteratively. At each step the values of the functionals are stored. Then these functional values are used for a curve fitting algorithm. The exact derivative of approximate curve is used as the third type of comparison sensitivity. All calculations have resulted in well agreements for the sensitivities. The difference between sensitivities are in the acceptable range (around.5 - 3. %). The procedure for the differential formulation however is easier than the integral formulation. It does not contain doubly nested integrals. The differential formulation does not need to calculate and use another fundamental solution. These advantages save computation time. The integral sensitivity analysis overcomes these disadvantages and it can give slightly better accuracies for sensitivities. On the other hand, choosing the formulation for the efficiency still depends on the experience of the user. The most important result of this study is the confirmation of the usability of the integral sensitivity analysis. For the systems which are described by integral equations, it is possible to use the integral sensitivity analysis by the adjoint variable method. xn
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