E.S. Mistakidis

On the approximation of nonmonotone multivalued problems by monotone subproblems

Abstract Structures involving zig-zag type reaction, displacement or stress-strain laws, i.e. nonmonotone possibly multivalued laws, cannot be effectively treated by the classical numerical methods for nonlinear laws. In order to calculate the arising free boundaries accurately, we propose a new approximation method of the nonmonotone problem by monotone ones. The proposed method finds its justification in the approximation of a hemivariational inequality by a sequence of variational inequalities. Also the case of combining this procedure with fixed point algorithms to treat more complicated problems is considered. The numerical methods proposed are illustrated mainly by numerical examples which are described in detail.

Low yield metal shear panels as an alternative for the seismic upgrading of concrete structures

The paper addresses the problem of seismic retrofitting of existing concrete structures through an innovative methodology based on low yield metallic shear panels. These panels are introduced in specific places in the concrete structure enhancing its strength and stiffness. However, the most important improvement concerns the energy dissipation capacity of the retrofitted structure. The paper presents a preliminary design methodology within the performance based design framework and details for the realization of the intervention. Elaborated non-linear finite element models are used in order to verify the quality of the results obtained from more simple models used in everyday engineering practice.

Steel T-Stub connections under static loading: an effective 2-D numerical model

Abstract The proposed numerical model concerns the simulation of the structural behaviour of steel bolted T-stub connections in the case where the development of zones of plastification, as well as unilateral contact effects on the interfaces between connection members and bolts, are taken into account. Within such a framework, an effective two-dimensional finite element model capable of describing plasticity, large displacement and unilateral contact effects is proposed. The model constitutes an easy-to-use and accurate numerical model for the analysis of steel connections subjected to tensile loading, and is a simplification of a respective three-dimensional one and aims to reduce in a reliable way the huge computational effort required for the analysis of fine meshes of discretized steel bolted connections. The validity of the assumptions that led to the proposed 2-D model is demonstrated by comparing the numerical results with those obtained by laboratory tests.

Fractal interfaces with unilateral contact and friction conditions

Abstract Structures involving interfaces with fractal geometry are analyzed here as a sequence of classical interface subproblems. On the interface, unilateral contact and friction boundary conditions are assumed to hold. These classical subproblems result from the consideration of the fractal interface as the ‘fixed point’ (or the ‘deterministic attractor’) of a given transformation. This approximation of the fractal is combined with a two-level contact-friction algorithm based on the optimization of the potential and of the complementary energy, after some appropriate transformations relying on the singular value decomposition of the equilibrium matrix are performed. Numerical examples illustrate the theory.

Comparison of two methods for the solution of a class of nonconvex energy problems using convex minimization algorithms

Nonmonotone, possibly multivalued stress-strain or reaction-displacement laws give rise to hemivariational inequalities. Due to the lack of convexity and the nonsmoothness of the underlying (super)potentials the problems generally have nonunique solutions (stable or unstable). In this paper we propose two methods for the solution of the hemivariational inequality problem. The first method is based on the decomposition of the nonconvex superpotential into convex constituents. The second one uses an iterative scheme in order to approximate the hemivariational inequality problem with a sequence of variational inequality problems. Both methods are based on the solution of convex subproblems and constitute an effective, reliable and versatile family of numerical algorithms for large scale hemivariational inequalities. Finally, the two methods are applied to solve the same problem and the obtained results are compared.

PARAMETRIC ANALYSIS OF THE STRUCTURAL RESPONSE OF STEEL BASE PLATE CONNECTIONS

Abstract The present paper aims to contribute to the parametric analysis of semirigid steel base plate connections. The method is based on the theoretical results of nonsmooth mechanics, which is a relatively new branch of mechanics initiated three decades ago and deals with problems from mechanics and/or structural analysis that involve generalizations of the gradient. The numerical modelling of the structural response of a steel column base plate under static loading where unilateral contact with friction and yielding phenomena are taken into account, could be considered as a typical problem that can be very effectively treated within such a theoretical framework. In particular, the respective stress states of the steel connection under static loading are calculated by taking into account the development of plastification zones and the unilateral contact and friction effects on the interfaces between connection members. An effective two-dimensional finite element model, capable of describing the previously mentioned phenomena is constructed. The aforementioned model is a simplification of the respective three-dimensional one and aims to reduce in a reliable way the huge computational effort required for the analysis of three-dimensional fine meshes of discretized steel connections. A numerical application demonstrates the effectiveness and the applicability of the method altering two parameters, the base plate thickness and the axial load of the model.

Numerical treatment of problems involving nonmonotone boundary or stress-strain laws

Abstract In order to describe the softening behavior of the materials, nonmonotone possible multivalued laws have been recently introduced. These laws are derived by nonconvex, generally nonsmooth energy functions called superpotentials that give rise to hemivariational inequalities. Due to the lack of convexity and the nonsmoothness of the underlying superpotentials these problems have generally nonunique solutions. On the other hand, problems involving monotone laws lead to variational inequalities that can be easily treated using modern convex minimization algorithms. The present paper proposes a new method for the solution of the nonmonotone problem by approximating it using monotone ones. The proposed method finds its justification in the approximation of a hemivariational inequality by a sequence of variational inequalities. This approach leads to effective reliable and versatile numerical alogrithms for large-scale hemivariational inequalities. The numerical method proposed is illustrated using appropriate examples.

On the numerical treatment of nonconvex energy problems of mechanics

The present paper presents three numerical methods devised for the solution of hemivariational inequality problems. The theory of hemivariational inequalities appeared as a development of variational inequalities, namely an extension foregoing the assumption of convexity that is essentially connected to the latter. The methods that follow partly constitute extensions of methods applied for the numerical solution of variational inequalities. All three of them actually use the solution of a central convex subproblem as their kernel. The use of well established techniques for the solution of the convex subproblems makes up an effective, reliable and versatile family of numerical algorithms for large scale problems. The first one is based on the decomposition of the contigent cone of the (super)-potential of the problem into convex components. The second one uses an iterative scheme in order to approximate the hemivariational inequality problem with a sequence of variational inequality problems. The third one is based on the fact that nonconvexity in mechanics is closely related to irreversible effects that affect the Hessian matrix of the respective (super)-potential. All three methods are applied to solve the same problem and the obtained results are compared.

Numerical treatment of hemivariational inequalities in mechanics : two methods based on the solution of convex subproblems

Hemivariational inequality problems describe equilibrium points (solutions) for structural systems in mechanics where nonmonotone, possibly multivalued laws or boundary conditions are involved. In the case of problems which admit a potential function this is a nonconvex, nondifferentiable one. In order to avoid the difficulties that arise during the calculation of equilibria for such mechanical systems, methods based on sequential convex approximations have recently been proposed and tested by the authors. The first method is based on ideas developed in the fields of quasidifferential and difference convex (d.c.) optimization and transforms the hemivariational inequality problem into a system of convex variational inequalities, which in turn leads to a multilevel (two-field) approximation technique for the numerical solution. The second method transforms the problem into a sequence of variational inequalities which approximates the nonmonotone problem by an iteratively defined sequence of monotone ones. Both methods lead to convex analysis subproblems and allow for treatment of large-scale nonconvex structural analysis applications.The two methods are compared in this paper with respect to both their theoretical assumptions and implications and their numerical implementation. The comparison is extended to a number of numerical examples which have been solved by both methods.

Friction evolution as a result of roughness in fractal interfaces

In this paper, the influence of fractal interface geometry to the evolution of the friction mechanism is studied. The paper is based on fractal approaches for the modeling of the multiscale self‐affine topography of these interfaces. More specifically, these approaches are based on scale‐independent parameters such as the fractal dimension. Here, friction between rough surfaces is assumed to be the result of the gradual plastification of the fractal interface asperities. In order to study the resulting highly nonlinear problem a variational formulation is used in order to describe contact between the interfaces. The numerical method used here leads to the successive solution of quadratic optimization problems. Finally, structures with different fractal interfaces are analyzed in order to obtain results for the relation between the fractal dimension and the overall response of the structures.