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This paper deals with the consideration of loading uncertainties in topology optimization via a fundamental optimization problem setting. Variability of loading in engineering design is realized e.g. in the action of various load combinations. In this study this phenomenon is modelled by the application of two mutually excluding (i.e. alternating) Optimization has since gained widespread application because efficiency and economical design are essential requirements in modern engineering. Continuous advance can be observed in several areas and applications in the industry not only in terms of practical design but also of fundamental research. The determination of the optimal structural topology is a branch of fundamental research, where the optimal topology is constructed in a design space based on given boundary conditions, serving as a decision support for actual design.The topology design historically started with the problem class of layout optimization of trusses and the works were called 'minimum volume design of frames' where the term 'frame' was historically used for what we now call a truss. The first important achievement in truss optimization was made by Maxwell [2], which deserves presentation here not only due to its significance but also because it is still unknown to many in the field. He considered the problem of attracting and repulsive forces between points set in the plane and proved a theorem regarding the equilibrium of the force system: 'In any system of points in equilibrium in a plane under the action of repulsions and attractions, the sum of the products of each attraction multiplied by the distance of the points between which it acts, is equal to the sum of the products of the repulsions multiplied each by the distance of the points between which it acts.' This statement can be formulated as t T t L t − c T c L c = 0 where T and L denote the force and the distance between two points, respectively, and indices t and c refer to tension (attraction) and compression (repulsion), respectively. For the derivation of this statement the principle of virtual work was applied with a uniform virtual deformation field.The far-reaching consequences of this theorem were not missed by Maxwell as he added therein: 'The importance of

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Topology optimization is complex engineering design tool. It needs intensive mathematical, mechanical and computing tools to perform the required design. During its hundred years of history it has become clear that the non-unique solution property of the method is affected by the material parameters (Poisson ratio) IntroductionThe engineering design is a very complex work. The designers have to take into consideration external (loading, design domain) and internal (effect of the numerical approximations) uncertain data and effect during this procedure. Sometimes the initial loading information has to recalculate (optimize) before the design [20] or due to the multiple solutions the designers have to select the most appropriate one. In engineering one can find an effective tool for these questions in topology optimization [15]. Topology optimization is one of the most popular parts of structural optimization. The "modern" period has been counted since the seminal paper of Bendsoe and Kikuchi in 1988 [4]. Topology optimization is a complex engineering design tool. It needs intensive mathematical, mechanical and computing tools to perform the required design. The method and the different solution techniques can be followed in several publications [1-3, 5, 7, 10, 15, 17, 19]. It has reached a rather high level of reputation in almost all field of life including many industrial fields and it has widespread academic use for structural optimization problems and also for material, mechanism, electromagnetics and other coupled field of design. Despite the level of research in topology optimization, several problems still exist concerning convergence, checkerboards and mesh-dependence which are subject to debate in the topology optimization community [6,8,9,[13][14][15][16]. During its hundred years of history it has become clear that the non-unique solution property of the method is affected by the material parameters (Poisson ratio) and the ways of the discretization. The applied finite element technique and the selected type of finite elements (generally four-nodes quadrilateral elements are used) can overcome numerical difficulties [9,[10][11][18][19]. From the very first start of the numerical solution technique of topology optimization, a serious problem with it was the erroneous appearance of corner contacts between solid elements in the solution (checkerboards, diagonal element chains, isolated hinges). To overcome this problem different techniques (some of them are heuristic) were applied [6, 8-10, 13-16, 18].

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