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EN
The paper presents a new methodology dedicated to design for additive manufacturing. Based on a hybrid algorithm of topological optimization, the method enables application of advanced constraints and validates solutions “on the fly” using virtual prototyping. Advanced constraints consider the influence of directions of additive manufacturing as well as the equivalent stress. In the optimization framework, real material properties related to three manufacturing directions were considered. The new design methodology is illustrated by benchmark tests and examples of wrist-hand orthosis topology optimization in which stress and manufacturing constraints were taken into account. As demonstrated by the conducted comparison tests with available commercial tools, the solutions obtained with the new method were characterized by lower mass and shorter computation time.
2
Content available remote Milestones in the 150-year history of topology optimization: a review
EN
Structural optimization is one of the most intensively investigated research areas in engineering. Recently, topology optimization has become the most popular engineering subfield. The starting date of structural optimization cannot be precisely determined. Michell’s optimization paper, published in 1904, is considered as the first publication in this subfield. However, his paper starts with a statement that his work is a generalization of Maxwell’s idea presented in the paper published in 1870. The authors of this review paper consider that this date can be accepted as the starting date of topology optimization. This paper is an overview of subjectively selected state-of-art achievements in topology optimization during its history of 150 years. The selection of the achievements is a rather difficult task because, in the early period of the history of topology optimization, a lot of meetings were classified and the results were not available for the public. The optimization community has almost no knowledge about the publications in topology optimization in the 1950s. Around that time, one can find some information on workshops and meetings connected to the Cambridge University or Oxford University with researchers such as Foulkes, Cox, Hemp, and Shield, who published significant results and these communications are generally not known for the reason mentioned above. After the 1970s, this situation has changed and there were more possibilities to find publications due to the changes and thanks to digitalization. As indicated earlier here subjectively selected works are overviewed from the 150-year history focusing on the first hundred twenty years.
EN
This paper analyzes the relations between the theory of Michell structures, which is one ofthe most important theories in structural optimization, and some remarkable engineeringstructures, including selected high-rise buildings, large-scale roof coverings and long-span bridges. The first part of this study briefly presents the development of Michell’s theory, its basic concepts, assumptions, and examples and fundamental features of Michellstructures. Then, several untypical engineering structures that make use of said conceptsare presented, including skyscrapers proposed by the Polish structural designer W. Zalewski and the international architectural office of Skidmore, Owings and Merill (SOM). Next, large-scale roof coverings of “Spodek” arena in Poland as well as selected bridgesare thoroughly analyzed in the context of similarity to Michell structures. The conductedstudy reveals that considered structural forms of the analyzed structures follow some ofthe concepts known from Michell’s theory and thus possess many features of the optimalstructural designs.
EN
A mathematical programming method to optimize the distribution field of a truss-like material is presented. The densities and angles of members are optimized in two separate procedures in each iteration. An explicit sub-problem in a variable separation form is established at every iteration procedure. At each sub-problem, the stress constraint function is expanded into a trigonometric series of the member angles. According to the extreme condition, the optimal orientations of members are determined. The member densities are optimized using the method of moving asymptotes (MMA). Two examples demonstrate that the optimal truss-like structures are very close to analytic solutions.
EN
This paper presents an application of topology optimization in bioengineering. The varia- tional approach to the topology optimization is applied. Using an original numerical algorithm and a programme developed in Matlab, the structure of the thighbone was modelled by FEM. The numerical results of the mass distribution in modelled bone are provided. Obtained topologies are similar to the density distribution in real bone tissue including the case when the implant is imposed into the human body.
PL
W artykule zaprezentowano nową metodę optymalizacji konstrukcji przy zastosowaniu optymalizacji topologicznej. Metoda pozwala uzyskać rozwiązania spełniające przyjęte ograniczenia technologiczne i wytrzymałościowe w pierwszym kroku bez konieczności żmudnego dostosowywania kształtu do postaci nadającej się do wytwarzania. Jak potwierdzają wyniki z przeprowadzonych testów, nowa metodyka umożliwia uzyskanie rozwiązań efektywnych, czyli lekkich i zwartych konstrukcji spełniających przyjęte ograniczenia i nadających do bezpośredniego wytwarzania technologia przyrostową.
EN
In the article, a new method for structural design using topological optimization is proposed. The method allows obtaining solutions that meet the adopted manufacturing and endurance constraints in the first step without the need of shape adjusting process to form suitable for production. As it is illustrated in the carried out tests, the proposed methodology allows to obtain effective solutions: lightweight and compact structures that meet the assumed constraints and give the possibility to direct fabrication by additive manufacturing.
7
Content available Optymalizacja topologiczna dźwigni pedału hamulca
PL
Przedstawiono sposób przeprowadzenia optymalizacji topologicznej dźwigni pedału hamulca w programie solidThinking Inspire. Pokazano geometrię przed optymalizacją, nadane warunki brzegowe optymalizacji oraz obliczeń wytrzymałościowych, a także geometrię po optymalizacji. Porównano wyniki obliczeń wytrzymałościowych oraz masy badanych obiektów przed i po optymalizacji. Zoptymalizowana geometria spełniła przyjęte kryteria przy jednoczesnym zmniejszeniu masy o 34,1%.
EN
The paper showcased topology optimization of the brake pedal lever performed in solidThinking Inspire software. Geometries before and after optimization were shown, as well as boundary conditions for optimization and strength assessment. The stresses and masses of geometries before and after optimization were compared. Optimized geometry met assumed strength criteria along with reduced mass by 34,1%.
8
Content available remote Weight reduction of motorcycle frame by topology optimization
EN
Purpose: of this paper is to improve the fuel efficiency of electrical motorcycle by reducing the weight of its frame without affecting the basic functionalities, dimensions and performance. Design/methodology/approach: Weight reduction of the frame was achieved by topology optimization technique. Initially the load and stresses acting on the frame was studied. Material of the frame was chosen as Aluminium and the frame was geometrically modelled using Autodesk Fusion 360. With the help of ANSYS AIM 18.2, weight of the frame was optimized by the design modifications suggested by the concept of topology optimization, for the corresponding loads and stresses induced on it. It was observed that the stress induced on the modified design was lesser than that of respective permissible yield stress of the frame material. After optimization, the weight of the frame was reduced from 3.0695 kg to 2.215 kg with the weight reduction of 27.84%. The weight reduction shows that the topology optimization is an effective technique, without compensate the performance of the frame. Approach used in the paper for the weight reduction of the frame is the topology optimization. The modelled frame was topology optimized by using ANSYS 18.2. After the topology optimization, the regions where the metal removal is possible, for weight reduction was identified. Findings: In this paper, the motor cycle frame was optimized and weight of the frame was reduced from 3.065 kg to 2.215 kg. Weight reduction of 27.84% was achieved without compensating the performance. Research limitations/implications: All the components of the automobile may be topology optimized for the weight reduction, thereby improving the fuel efficiency. Innovative design/Improvement in design also possible. Practical implications: By reducing the weight of the frame, weight of the automobile also reduces. Reduction in weight of the automobile leads to improved fuel efficiency. Originality/value: Weight of the motorcycle frame reduced by topology optimization. The regions of material removal at the frame, without compensating the performance was identified.
EN
This paper presents a dynamic analysis of earlier optimized auxetic structure. This optimization based on the distribution of two materials in such way to obtain a minimal value of Poisson’s ratio (PR), which indicates the auxetic properties. The initial optimized shape was so-called star structure, which if is made from one material has the PR close to 0.188. After optimization with the goal function of PR-minimization, the obtained value was equal to -9.5043. Then the eigenfrequencies for the optimized structure were investigated. The calculations were carried out by means of Finite Element Method (FEM). For optimization of the value of Poisson’s ratio was used algorithm MMA (Method of Moving Asymptotes). The computing of single material properties (PR, Young’s modulus, density) for the whole shape was made by means of SIMP method (Solid Isotropic Method with Penalization).
PL
Przedstawiono zastosowanie tzw. optymalizacji topologicznej do wyrobów produkowanych poprzez wtryskiwanie tworzyw sztucznych. Jest to metoda obliczeniowa pozwalająca na uzyskanie maksymalnej sztywności geometrii wypraski, a jednocześnie redukcję jej masy. Zaprezentowano możliwości wybranych narzędzi numerycznych wykorzystujących optymalizację topologiczną i ograniczenia w kontekście procesu wtryskiwania.
EN
The use of so-called topological optimization for products manufactured by means of injection molding of plastics, is presented. This is a computational method that allows for maximum rigidity of the mold geometry, while reducing its mass. Possibilities of selected numerical tools using topological optimization and constraints in the context of injection molding process, are presented.
11
Content available remote Simultaneous design of optimal shape and local cubic material characteristics
EN
This paper deals with the minimum compliance problem of the femur bone made of a nonhomogeneous elastic material with cubic symmetry. The elastic moduli as well as the trajectories of anisotropy directions are design variables. The isoperimetric condition determines the value of the cost of the design expressed as the integral of the trace of the Hooke’s tensor. The optimum design is found for a selected design domain and a single load case. The optimal cubic material characteristics are reflected by the properties of the underlying microstructure. Admissible microstructures are reconstructed, thus delivering a deeper insight into the optimum design. The obtained microstructures are second- rank laminates composed of an isotropic material and voids. To eliminate the degeneracy of the design at least three load cases should be considered.
12
Content available remote Efficient generator of structural topologies based on irregular cellular automata
EN
Recent development of Cellular Automata implementation into optimal design problems has shown that the automaton can be an effective tool for generation of optimal topologies in engineering applications. Nevertheless, the vast majority of results have been obtained to date for regular lattices of cells. The aim of the present paper is therefore to extend the concept of Cellular Automata towards irregular grid of cells related to non-regular mesh of finite elements. Introducing irregular lattice of cells allows to reduce number of design variables without losing accuracy of results and without excessive increase of number of elements caused by using fine mesh for a whole structure. This paper proposes a novel Irregular Cellular Automata formulation that can be adapted to topology optimization of real structural elements. The effectiveness of proposed local update rule is illustrated by results of numerical generation of optimal topologies for selected spatial engineering structures.
EN
The hybrid structures show excellent performance on vibration reduction for ship, aircraft and spacecraft designs. Meanwhile, the topology optimization is widely used for structure vibration reduction and weight control. The design of hybrid structures considering simultaneous materials selection and topology optimization are big challenges in theoretical study and engineering applications. In this paper, according to the proposed laminate component method (LCM) and solid isotropic microstructure with penalty (SIMP) method, the mathematical formulations are presented for concurrent materials selection and topology optimizations of hybrid structures. Thickness distributions of the plies in laminate components are defined as materials selection design variables by LCM method. Relative densities of elements in the components are defined as topology design variables by SIMP method. Design examples of hybrid 3-bar truss structures and hybrid floating raft with vibration reduction requirements verified the effectiveness of the presented optimization models.
EN
The present paper delivers some new numerical and exact solutions of the three forces problem that is one of fundamental problems of Michell’s truss theory. The present problem is to find the lightest fully stressed truss transmitting three self-equilibrated co-planar forces. In this study, we limit our considerations to the case of two forces being mutually orthogonal. The aim of this paper is to classify possible layouts of optimal trusses depending on the position of the applied lateral point load (the positions of the other two forces are fixed, which however does not restrict the scope of the study). The exact analytical solutions are obtained with a great help of numerical solutions that enable proper prediction of the optimal layouts.
15
Content available remote A novel heuristic algorithm for minimum compliance topology optimization
EN
The implementation of efficient and versatile methods to the generation of optimal topologies for engineering structural elements is one of the most important issues stimulating progress within the structural topology optimization area. Over the years, optimization problems have been typically solved by the use of classical gradient-based mathematical programming algorithms. Nowadays, these traditional techniques are more often replaced by other algorithms, usually by the ones based on heuristic rules. Heuristic optimization techniques are gaining widespread popularity among researchers because they are easy to implement numerically, do not require gradient information, and one can easily combine this type of algorithm with any finite element structural analysis code. In this paper, a novel heuristic algorithm for a minimum compliance topology optimization is proposed. Its effectiveness is illustrated by the results of numerical generation of optimal topologies for selected plane structures.
PL
W artykule omówiono istotę i cel optymalizacji topologicznej. Dokonano przeglądu tradycyjnych rozwiązań konstrukcyjnych stosowanych w połączeniach trzonu słupa oświetleniowego z blachą podstawy. Przedstawiono wyniki autorskich badań z zakresu optymalizacji połączenia słup-blacha podstawy. Omówiono różnice między rozwiązaniami uzyskanymi w drodze optymalizacji topologicznej a rozwiązaniami tradycyjnymi.
EN
The article discusses the nature and purpose of the topology optimization. An overview of current construction solutions used in the connections column-base plate was done. We also present the results of our own research in the field of optimization of the connection column-base plate. The differences between the solutions obtained through topology optimization and traditional solutions was discussed.
PL
W artykule przedstawiono porównanie dwóch metod optymalizacji korpusów maszyn. Obie metody wykorzystują zarówno metodę elementów skończonych jak i algorytm ewolucyjny. Pierwsza z nich zakłada, że znany jest model wstępny obiektu i wówczas należy użyć tylko optymalizacji parametrycznej. Natomiast w drugim przypadku, kiedy nie ma żadnych informacji o modelu obiektu należy zastosować zarówno optymalizacje topologiczną jak i parametryczną. Ta druga metoda wykorzystuje prymitywy, jako modele wstępne obiektu. W artykule zamieszczono wyniki porównania obu metod dla wybranego korpusu obrabiarki. Porównaniu podlegały optymalne rozwiązania w postaci: wskaźników sztywności, ich rozrzutu, masy korpusów i częstotliwość drgań własnych. Wyniki tych porównań są dosyć oczywiste: metoda optymalizacji, bazująca na prymitywach daje korzystniejsze efekty niż metoda bazująca na projekcie wstępnym. Dotyczy to w szczególności masy zoptymalizowanego korpusu, która może być nawet o 10 mniejsza.
EN
The paper presents comparison of two optimisation methods of machine frames. Both methods use Finite Element Methods and Evolutionary Algorithm simultanously. The first of the method assumes that the initail model of the body is known and in such situation the parametric optimisation should be applied only. In the second case when one has no information about the object’s model, the Topology optimisation and Parametric optimisation should be applied. The second method uses Prymitives as preliminary model of object. The paper presents results of comparision of both metof applied to an example frame. Such parameters were compared: coefficients of stiffness, dispersion of stiffness, masses od frames, free frequency of vibration. Results of comparision are very obviousness: method of optimisation based on primitives gives better results than method based on initialy project. First of all it concerns on the mass of opimised frame, which may be even 10% smaller.
EN
In this paper, a new hybrid evolutionary-gradient algorithm is proposed to solve global numerical problems of topology optimization. The proposed algorithm is developed as a compromise between the direct and gradient-based optimization approaches. A short comparative study of computational results demonstrates the validity of the proposed method, as well as the necessity of its use.
PL
W niniejszej pracy zaproponowano nowy, hybrydowy algorytm gradientowo-ewolucyjny do numerycznego rozwiązywania zagadnień z zakresu globalnej optymalizacji topologii. Proponowany algorytm stanowi kompromis między bezpośrednimi i gradientowymi metodami optymalizacji. Krótka analiza porównawcza wyników obliczeń wykazuje walidację proponowanej metody, a także konieczność jej stosowania.
19
Content available remote Isotropic Material Design
EN
The paper deals with optimal distribution of the bulk and shear moduli minimizing the compliance of an inhomogeneous isotropic elastic 3D body transmitting a given surface loading to a given support. The isoperimetric condition is expressed by the integral of the trace of the Hooke tensor being a linear combination of both moduli. The problem thus formulated is reduced to an auxiliary 3D problem of minimization of a certain stress functional over the stresses being statically admissible. The integrand of the auxiliary functional is a linear combination of the absolute value of the trace and norm of the deviator of the stress field. Thus the integrand is of linear growth. The auxiliary problem is solved numerically by introducing element-wise polynomial approximations of the components of the trial stress fields and imposing satisfaction of the variational equilibrium equations. The under-determinate system of these equations is solved numerically thus reducing the auxiliary problem to an unconstrained problem of nonlinear programming.
EN
Distribution optimization of elastic material under elastic isotropic rectangular thin plate subjected to concentrated moving load is investigated in the present paper. The aim of optimization is to damp its vibrations in finite (fixed) time. Accepting Kirchhoff hypothesis with respect to the plate and Winkler hypothesis with respect to the base, the mathematical model of the problem is constructed as two–dimensional bilinear equation, i.e. linear in state and control function. The maximal quantity of the base material is taken as optimality criterion to be minimized. The Fourier distributional transform and the Bubnov–Galerkin procedures are used to reduce the problem to integral equality type constraints. The explicit solution in terms of two–dimensional Heaviside‘s function is obtained, describing piecewise–continuous distribution of the material. The determination of the switching points is reduced to a problem of nonlinear programming. Data from numerical analysis are presented.
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