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1

On the robustness of the topological derivative for Helmholtz problems and applications

Leugering Günter, Novotny Antonio André, Sokolowski Jan

Control and Cybernetics

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2022

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Vol. 51, No. 2

227--248

EN

We consider Helmholtz problems in two and three dimensions. The topological sensitivity of a given cost function J(uɛ) with respect to a small hole Bɛ around a given point x0ɛ ∈ Bɛ ⊂ Ω depends on various parameters, like the frequency k chosen or certain material parameters or even the shape parameters of the hole Bɛ. These parameters are either deliberately chosen in a certain range, as, e.g., the frequencies, or are known only up to some bounds. The problem arises as to whether one can obtain a uniform design using the topological gradient. We show that for 2-d and 3-d Helmholtz problems such a robust design is achievable.

2

Structure of Optimal Control in Optimal Shaping of the Steel Arch

Mikulski Leszek, Jasińska Dorota, Dąbrowska Olga

Civil and Environmental Engineering Reports

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2022

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Vol. 32, no. 3

143--165

EN

The paper presents the problem of optimal shaping of the H-bar cross-section of a steel arch that ensures minimal mass. Nineteen combinations of nine basic load states are considered simultaneously in the problem formulation. The optimal shaping task is formulated as a control theory problem within the formal structure of the maximum Pontriagin’s principle. Since the ranges of constraint activity defining the control structure are a priori unknown and must be determined numerically, assuming the proper control structure plays a key role in the task solution. The main achievement of the present work is the determination of a solution of the multi-decision and multi-constraint optimization problem of the arch constituting a primary structural system of the existing building assuring the reduction of the structure mass up to 42%. In addition, the impact of the assumed state constraint value on the solution structure is examined.

3

The stress-minimizing hole in a shear-loaded elastic plate at a given energy increment

Vigdergauz S.

Archives of Mechanics

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2022

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Vol. 74, nr 2-3

109--126

EN

Minimization of the peak tangential stresses around a single hole in an infinite 2D elastic plate under remote pure shear and a given hole-induced strain energy level is considered as a free-shape optimization problem under a physical constraint. It is solved by combining a genetic algorithm with the almost analytical, and hence highly accurate stress-strain solver for any finitely parameterized family of closed curves. The results obtained in wide ranges of the governing parameters are detailed and discussed. They may be applicable to the optimal holes design in constructive elements and dilute perforated structures. The current analysis extends the author’s previous publications, which were focused on the unconstrained shape optimization within the same setup.

4

Trójparametryczna optymalizacja trzonu pełnościennego słupa energetycznego

Wandzik Grzegorz, Potempa Rafał

Inżynieria i Budownictwo

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2021

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R. 77, nr 5-6

195--199

PL

Przedstawiono metodę poszukiwania optymalnego kształtu stalowego słupa energetycznego o konstrukcji powłokowej. Optymalizacja została ograniczona do trzech parametrów: średnicy, zbieżności trzonu, grubości ścianki powłoki. Podstawą analiz jest minimalizacja objętości stali, z użyciem dostosowanej do problemu metody gradientu prostego. Algorytm zastosowano w autorskim programie komputerowym.

EN

The method of searching the optimal shape of a steel high-voltage thin-shell pole was presented. The optimization was limited to three parameters: the diameter, convergence of the shaft, the wall thickness of the shell. The basis of the analysis is the minimization of the steel volume of the pole shaft using the simple gradient descent method adapted to the problem. The algorithm has been implemented in a proprietary computer application.

5

Simultaneous shape and mesh quality optimization using pre-shape calculus

Luft Daniel, Schulz Volker

Control and Cybernetics

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2021

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Vol. 50, No. 4

473--520

EN

Computational meshes arising from shape optimization routines commonly suffer from decrease of mesh quality or even destruction of the mesh. In this work, we provide an approach to regularize general shape optimization problems to increase both shape and volume mesh quality. For this, we employ pre-shape calculus as established in Luft and Schulz (2021). Existence of regularized solutions is guaranteed. Further, consistency of modified pre-shape gradient systems is established. We present pre-shape gradient system modifications, which permit simultaneous shape optimization with mesh quality improvement. Optimal shapes to the original problem are left invariant under regularization. The computational burden of our approach is limited, since additional solution of possibly larger (non-)linear systems for regularized shape gradients is not necessary. We implement and compare pre-shape gradient regularization approaches for a 2D problem, which is prone to mesh degeneration. As our approach does not depend on the choice of metrics representing shape gradients, we employ and compare several different metrics.

6

Pre-shape calculus and its application to mesh quality optimization

Luft Daniel, Schulz Volker

Control and Cybernetics

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2021

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Vol. 50, No. 3

263--301

EN

Deformations of the computational mesh, arising from optimization routines, usually lead to decrease of mesh quality or even destruction of the mesh. We propose a theoretical framework using pre-shapes to generalize the classical shape optimization and calculus. We define pre-shape derivatives and derive corresponding structure and calculus theorems. In particular, tangential directions are featured in pre-shape derivatives, in contrast to classical shape derivatives, featuring only normal directions. Techniques from classical shape optimization and calculus are shown to carry over to this framework. An optimization problem class for mesh quality is introduced, which is solvable with the use of pre-shape derivatives. This class allows for simultaneous optimization of the classical shape objectives and mesh quality without deteriorating the classical shape optimization solution. The new techniques are implemented and numerically tested for 2D and 3D.

7

Optimization of material distribution for forged automotive components using hybrid optimization techniques

Sebastjan Przemysław, Kuś Wacław

Computer Methods in Materials Science

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2021

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Vol. 21, No. 2

63--74

EN

The paper deals with the problem of optimal material distribution inside the provided design area. Optimization based on deterministic and stochastic algorithms is used to obtain the best result on the basis of the proposed objective function and constraints. The optimization of the shock absorber is used as an example of the described methods. One of the main difficulties addressed is the manufacturability of the optimized part intended for the forging process. Additionally, nonlinear buckling simulation with the use of the finite element method is used to solve the misuse case of shock absorber compression, where the shape of the optimized part has a key role in the total strength of the automotive damper. All of that, together with the required design precision, creates the nontrivial constrained optimization problem solved using the parametric, implicit geometry representation and a combination of stochastic and deterministic algorithms used with parallel design processing. Two methods of optimization are examined and compared in terms of the total amount of function calls, final design mass, and feasibility of the resultant design. Also, the amount of parameters used for the implicit geometry representation is greatly reduced compared to existing schemes presented in the literature. The problem addressed in this article is strongly inspired by the actual industrial example of the mass minimization process, but it is more focused on the actual manufacturability of the resultant component and admissible solving time. Commercially accessible software combined with authors’ procedures is used to resolve the material distribution task, which makes the proposed method universal and easily adapted to other fields of the optimization of mechanical elements.

8

Designing of Steel CHS Columns Showing Maximum Compression Resistance

Marcinowski Jakub, Sadowski Mirosław

Civil and Environmental Engineering Reports

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2021

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Vol. 31, no. 1

79--92

EN

The paper deals with a shape optimisation procedure of steel, compressed bars. Circular hollow sections (CHS) of variable cross sections and variable wall thickness are taken into account. The proposed procedure for designing of steel rods exhibiting maximum compression resistance is effective and possible to use in engineering practice. The advantage of the proposed shape of the bar is that it allows to increase the value of its load carrying capacity, i.e. it ensures the transfer of a higher value of compressive force than similar, solid struts of the same mass and length. The extent of the increase in the load capacity relative to the load capacity of the reference solid, cylindrical bar depends on the slenderness of the reference bar and ranges from 60% to 170%. Due to this very beneficial fact, it can be used wherever it is required to maintain a certain stiffness and an increased value of compressive force is desired, as well as in constructions where it is necessary to reduce weight while maintaining the adopted mechanical parameters, e.g. values of load bearing capacity. Final results achieved in the research were presented in the form of the flow chart allowing to design the compressed columns of optimum shape.

9

Using the Erfi function in the problem of the shape optimization of the compressed rod

Marcinowski J., Sadowski M.

International Journal of Applied Mechanics and Engineering

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2020

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Vol. 25, no. 2

75--87

EN

The shape of the optimal rod determined in the work meets the condition of mass conservation in relation to the reference rod. At the same time, this rod shows a significant increase in resistance to axial force. In the examples presented, this increase was 80% and 117%, respectively, for rods with slenderness of 125 and 175. A practical benefit from the use of compression rods of the proposed shapes is clearly visible. The example presented in this publication shows how great the utility in the structural mechanics can be, resulting from the applications of complex analysis (complex numbers). This approach to many problems can find its solutions, while they are lacking in the real numbers domains. What is more, although these are operations on complex numbers, these solutions have often their real representations, as the numerical example shows. There are too few applications of complex numbers in the technique and science, therefore it is obvious that the use of complex analysis should have an increasing range. One of the first people to use complex numbers was Girolamo Cardano. Cardano, using complex numbers, was solving cubic equations, unsolvable to his times – as the famous Franciscan and professor of mathematics Luca Pacioli put it in his paper Summa de arithmetica, geometria, proportioni et proportionalita (1494). It is worth mentioning that history has given Cardano priority in the use of complex numbers, but most probably they were discovered by another professor of mathematics – Scipione del Ferro (cf. [1]). We can see, that already then, they were definitely important (complex numbers).

10

Shape sensitivity of optimal control for the Stokes problem

Abdelbari Merwan, Nachi Khadra, Sokolowski Jan, Szulc Katarzyna

Control and Cybernetics

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2020

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Vol. 49, No. 1

11--40

EN

In this article, we study the shape sensitivity of optimal control for the steady Stokes problem. The main goal is to obtain a robust representation for the derivatives of optimal solution with respect to smooth deformation of the flow domain. We introduce in this paper a rigorous proof of existence of the material derivative in the sense of Piola, as well as the shape derivative for the solution of the optimality system. We apply these results to derive the formulae for the shape gradient of the cost functional; under some regularity conditions the shape gradient is given according to the structure theorem by a function supported on the moving boundary, then the numerical methods for shape optimization can be applied in order to solve the associated optimization problems.

11

Size and shape design optimization of truss structures using the Jaya algorithm

Grzywiński Maksym

Computer Assisted Methods in Engineering and Science

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2020

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Vol. 27, no. 2-3

177--184

EN

The metaheuristic algorithm is proposed to solve the weight minimization problem of trussstructures, considering the shape and sizing design variables. Design variables are discreteand/or continuous. The design of truss structures is optimized by an efficient optimiza-tion algorithm called Jaya. The main feature of Jaya is that it does not require settingalgorithm-specific parameters. The algorithm has a very simple formulation in which thebasic idea is to approach the best solution and escape from the worst solution [6]. Analysesof structures are performed by a finite element code in MATLAB. The effectiveness of theJaya algorithm is demonstrated using two benchmark examples: planar truss 18-bar andspatial truss 39-bar, and compared with results in references.

12

Weight minimization of spatial trusses with genetic algorithm

Grzywiński Maksym, Selejdak Jacek

Quality Production Improvement - QPI

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2019

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Vol. 1, Iss. 1

238--243

EN

A genetic algorithm is proposed to solve the weight minimization problem of spatial truss structures considering size and shape design variables. A very recently developed metaheuristic method called JAYA algorithm (JA) is implemented in this study for optimization of truss structures. The main feature of JA is that it does not require setting algorithm specific parameters. The algorithm has a very simple formulation where the basic idea is to approach the best solution and escape from the worst solution. Analyses of structures are performed by a finite element code in MATLAB. The effectiveness of JA algorithm is demonstrated through benchmark spatial truss 39-bar, and compare with results in references.

13

Shape optimization of a column under generalized load with a force directed towards positive pole

Szmidla Janusz, Jurczyńska Anna

Quality Production Improvement - QPI

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2019

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Vol. 1, Iss. 1

326--333

EN

In this paper, the issue of shape optimization of a column subjected to the generalized load with a force directed towards the positive pole (L. Tomski’s load, specific load) was considered. Based on the Hamilton’s principle, the differential equations of movement and boundary conditions describing the system were formulated. Taking into account a kinetic criterion of stability loss and a condition of constant total volume, the scope of changes in natural frequency as a function of an external load was determined with selected geometrical and physical parameters of the loading structure. On the basis of obtained results, values of geometrical parameters of individual column segments were determined, at which the maximum critical load value was obtained. In order to find the maximum critical force, which is a function of many variables, the simulated annealing algorithm was used.

14

Optimal heat distributions by a gradient-based shape optimization method

Belhachmi Z., Meftahi B., Meftahi H.

Control and Cybernetics

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2018

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Vol. 47, no. 1

33--53

EN

In this paper, we consider the problem of locating coated inclusions in a 2D dimensional conductor material in order to obtain a suitable thermal environment. The mathematical model is described by elliptic partial diﬀerential equation with linear boundary condition, including heat transfer coeﬃcient. A shape optimization problem is formulated by introducing a cost functional to solve the problem under consideration. The shape sensitivity analysis is rigorously performed for the problem by means of a Lagrangian formulation. The optimization problem is solved by means of gradient-based strategy and numerical experiments are carried out to demonstrate the feasibility of the approach.

15

Design and Strength Analysis of Curved-Root Concept for Compressor Rotor Blade in Gas Turbine

Wdowiński W., Szymczyk E., Jachimowicz J., Moneta G.

Fatigue of Aircraft Structures

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2017

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Iss. 9

137--155

EN

The motivation of the article is fatigue and fretting issue of the compressor rotor blades and disks. These phenomena can be caused by high contact pressures leading to fretting occurring on contact faces in the lock (blade-disk connection, attachment of the blade to the disk). Additionally, geometrical notches and high cyclic loading can initiate cracks and lead to engine failures. The paper presents finite element static and modal analyses of the axial compressor 3rd for the original trapezoidal/dovetail lock geometry and its two modifications (new lock concepts) to optimize the stress state of the disk-blade assembly. The cyclic symmetry formulation was used to reduce modelling and computational effort.

16

Topological sensitivity analysis for a coupled nonlinear problem with an obstacle

Abdelbari M., Nachi K., Sokolowski J.

Control and Cybernetics

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2017

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Vol. 46, no. 1

5--25

EN

The Topological Derivative has been recognized as a powerful tool in obtaining the optimal topology for several kinds of engineering problems. This derivative provides the sensitivity of the cost functional for a boundary value problem for nucleation of a small hole or a small inclusion at a given point of the domain of integration. In this paper, we present a topological asymptotic analysis with respect to the size of singular domain perturbation for a coupled nonlinear PDEs system with an obstacle on the boundary. The domain decomposition method, referring to the SteklovPoincar´epseudo-diﬀerential operator, is employed for the asymptotic study of boundary value problem with respect to the size of singular domain perturbation. The method is based on the observation that the known expansion of the energy functional in the ring coincides with the expansion of the Steklov-Poincar´e operator on the boundary of the truncated domain with respekt to the small parameter, which measures the size of perturbation. In this way, the singular perturbation of the domain is reduced to the regular perturbation of the Steklov-Poincar´e map ping for the ring. The topological derivative for a tracking type shape functional is evaluated so as to obtain the useful formula for application in the numerical methods of shape and topology optimization.

17

Shape optimization of road tunnel cross-section by simulated annealing

Sobótka M., Pachnicz M.

Studia Geotechnica et Mechanica

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2016

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Vol. 38, nr 2

47--52

EN

The paper concerns shape optimization of a tunnel excavation cross-section. The study incorporates optimization procedure of the simulated annealing (SA). The form of a cost function derives from the energetic optimality condition, formulated in the authors’ previous papers. The utilized algorithm takes advantage of the optimization procedure already published by the authors. Unlike other approaches presented in literature, the one introduced in this paper takes into consideration a practical requirement of preserving fixed clearance gauge. Itasca Flac software is utilized in numerical examples. The optimal excavation shapes are determined for five different in situ stress ratios. This factor significantly affects the optimal topology of excavation. The resulting shapes are elongated in the direction of a principal stress greater value. Moreover, the obtained optimal shapes have smooth contours circumscribing the gauge.

18

Optimal reliability for components under thermomechanical cyclic loading

Bittner L., Gottschalk H.

Control and Cybernetics

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2016

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Vol. 45, no. 4

421--455

EN

We consider the existence of optimal shapes in a context of the thermo-mechanical system of partial diﬀerential equations (PDE) using the recent approach based on elliptic regularity theory (Gottschalk and Schmitz, 2015; Agmon, Douglis and Nirenberg, 1959,1964; Gilbarg and Trudinger, 1977). We give an extended and improved deﬁnition of the set of admissible shapes based on a class of suﬃciently diﬀerentiable deformation maps applied to a baseline shape. The obtained set of admissible shapes again allows to prove a uniform Schauder estimate for the elasticity PDE. In order to deal with thermal stress, a related uniform Schauder estimate will be derived for the heat equation. Special emphasis is put on Robin boundary conditions, which are motivated by the convective heat transfer processes. It is shown that these thermal Schauder estimates can serve as an input to the Schauder estimates for the elasticity equation (Gottschalk and Schmitz, 2015). This is needed to prove the compactness of the (suitably extended) solutions of the entire PDE system in some state space that carries a C2-Hölder topology for the temperature ﬁeld and a C3-Hölder topology for the displacement. From this, one obtains the property of graph compactness, which is the essential tool to prove the existence of optimal shapes. Due to the topologies employed, the method works for objective functionals that depend on the displacement and its derivatives up to third order, as well as on the temperature ﬁeld and its derivatives up to second order. This general result in shape optimization is then applied to the problem of optimal reliability, i.e. the problem of ﬁnding shapes that have minimal failure probability under cyclic thermomechanical loading.

19

Shape optimization of the modular press body

Pabiszczak S.

Archives of Mechanical Technology and Materials

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2016

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Vol. 36

59--64

EN

A paper contains an optimization algorithm of cross-sectional dimensions of a modular press body for the minimum mass criterion. Parameters of the wall thickness and the angle of their inclination relative to the base of section are assumed as the decision variables. The overall dimensions are treated as a constant. The optimal values of parameters were calculated using numerical method of the tool Solver in the program Microsoft Excel. The results of the optimization procedure helped reduce body weight by 27% while maintaining the required rigidity of the body.

20

Optymalizacja kształtu odkuwki kołnierzowej z wykorzystaniem metody Rosenbrocka w celu minimalizacji jej masy

Gronostajski Z., Hawryluk M., Misiun G., Polak S., Pawełczyk M.

Prace Naukowe Politechniki Warszawskiej. Mechanika

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2015

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z. 267

97--102

PL

W pracy autorzy podjęli się optymalizacji odkuwki kołnierzowej w celu minimalizacji masy materiału wsadowego. Przemysłowy proces kucia realizowany jest na prasie korbowej o dopuszczalnym nacisku 2500 ton w 3 operacjach: spęczania, kucia matrycowego wstępnego i kucia matrycowego wykańczającego. Do przeprowadzonej optymalizacji wykorzystano bezgradientowy algorytm Rosenbrocka, który sprzężono z modelem numerycznym przy wykorzystaniu języka Python. W ramach badań przeprowadzono modelowanie numeryczne procesu kucia, które pozwoliło na określenie jego najistotniejszych parametrów, m.in.: sił, rozkładów odkształcenia i pola temperatur oraz sposobu płynięcia. Główne parametry optymizacji dobrano na podstawie wyników symulacji komputerowych oraz normy kucia odkuwki kołnierzy. Przyjęto, że minimalizacja masy odkuwki będzie realizowana poprzez zmianę trzech wybranych na podstawie wyników MES, najistotniejszych parametrów – grubości i położenia denka oraz wysokości otwarcia na wypływkę. Wynikiem przeprowadzonej optymalizacji było obniżenie masy materiału o 5,3%. Otrzymane rezultaty optymalizacji zostały poddane analizie a następnie weryfikacji opartej o eksperyment w warunkach przemysłowych.

EN

In presented article authors has decided to optimize forging of the flange type in order to minimize the weight of the initial billet. Industrial forging process was carried out on the crank press with the pressure of 2500 tonnes in three operations: upsetting, initial die forging and final die forging. To conduct the optimization Rosenbrock algorithm has been used. For the study numerical modeling of industrial forging process has been carried out. Based on the results of computer simulation and standards of flange forging authors decided to reduce the weight by interfering into three selected parameters - thickness and the position of the bottom and the height of the outside flash opening. The result of the optimization was material weight reduction by 5,3% The obtained results were analyzed and will be verified by experiment in industrial conditions.