We consider the fourth-order elliptic equation describing the Kirchhoff-Love model for pure bending of a thin solid symmetric plate under a transverse load. Following the results of Antonić and ...Balenović (1999, 2000) 4,5, and Burazin, Jankov and Vrdoljak (2018) 12, we show the local character of the set of all possible composites, also called the G-closure, and prove that the set of composites obtained by periodic homogenization is dense in that set. Moreover, we derive expressions for elastic coefficients of composite plate obtained by mixing two materials in thin layers, also known as laminated materials, and for mixing two materials in the low-contrast regime. Additionally, we derive optimal bounds on the effective energy of a composite material, known as Hashin-Shtrikman bounds. In the setting of two-phase isotropic materials in dimension d=2, explicit optimal Hashin-Shtrikman bounds are calculated.
We consider multiple state optimal design problems with two isotropic materials from the conductivity point of view. Since the classical solutions of these problems usually do not exist, a proper ...relaxation of the original problem is obtained, using the homogenization method. In 1 we derive necessary conditions of optimality of the relaxed problem, which enables us to implement a new variant of the optimality criteria method. It appears that this variant gives converging sequence of designs for the energy minimization problems. In this work we prove convergence of the method for energy minimization problems in the spherically symmetric case and in a case when the number of states is less than the space dimension.
A method for the optimization of the distribution of two materials in a thin symmetric plate is presented. The model for the plate is the Kirchhoff–Love and the optimization method is based on the ...small amplitude, or low contrast, approximation for homogenization. Numerical results are presented for compliance minimization, i.e., maximizing the global stiffness of the plate, and for minimizing the integral of an even power of the deflection in a subset of the plate. The method arrives at classical designs without the need of penalization, which allows evaluating the exact performance of the proposed shapes and the comparison with some simple intuitive designs. This also gives insight in how small the contrast needs to be for the method to give designs which are better than some intuitive designs.
Evolutionary equations are G-compact Burazin, Krešimir; Erceg, Marko; Waurick, Marcus
Journal of evolution equations,
06/2024, Letnik:
24, Številka:
2
Journal Article
Recenzirano
Odprti dostop
We prove a compactness result related to
G
-convergence for autonomous evolutionary equations in the sense of Picard. Compared to previous work related to applications, we do not require any ...boundedness or regularity of the underlying spatial domain; nor do we assume any periodicity or ergodicity assumption on the potentially oscillatory part. In terms of abstract evolutionary equations, we remove any compactness assumptions of the resolvent modulo kernel of the spatial operator. To achieve the results, we introduced a slightly more general class of material laws. As a by-product, we also provide a criterion for
G
-convergence for time-dependent equations solely in terms of static equations.
In this paper we explicitly calculate the first correction in the small-amplitude homogenization limit of a sequence of periodic tensors describing material properties in the Kirchhoff model for pure ...bending of a thin solid symmetric plate under a transverse load. We also discuss smooth dependence of H-limit on a parameter and calculate H-limit of a periodic sequence of tensors.
In optimal design problems the goal is to find an optimal distribution of materials in a fixed domain which optimizes a criterion, computed through the solution of a partial differential equation ...modelling the involved physics. We consider optimal design problems in the setting of linearized elasticity, and restrict ourselves to domains filled with two isotropic elastic phases. Since a classical solution usually does not exist, we use the homogenization method in order to get a proper relaxation of the original problem. Unfortunately, the related G-closure problem in linearized elasticity is not solved. However, in the compliance minimization problem, G-closure can be replaced by a smaller subset made of sequential laminates, which is explicitly known. Moreover, for the compliance minimization the necessary conditions of optimality are easily derived, which enables a development of optimality criteria method for the numerical solution. We explicitly calculate the lower Hashin-Shtrikman bound on the complementary energy in two space dimensions, and necessary conditions of optimality. We develop an optimality criteria method for the single state compliance minimization problems.
We study the uniqueness and explicit derivation of the relaxed optimal solutions, corresponding to the minimization of weighted sum of potential energies for a mixture of two isotropic conductive ...materials on an annulus. Recently, it has been shown by Burazin and Vrdoljak that even for multiple-state problems, if the domain is spherically symmetric, then the proper relaxation of the problem by the homogenization method is equivalent to a simpler relaxed problem, stated only in terms of local proportions of given materials. This enabled explicit calculation of a solution on a ball, while problems on an annulus appeared to be more tedious. In this paper, we discuss the uniqueness of a solution of this simpler relaxed problem, when the domain is an annulus and we use the necessary and sufficient conditions of optimality to present a method for explicit calculation of the unique solution of this simpler proper relaxation, which is demonstrated on an example.
Small‐amplitude homogenization is a procedure for computing the first correction in the homogenization limit. By using H‐measures, we calculate the first correction in the small‐amplitude ...homogenization limit of a non‐periodic sequence of tensors describing material properties in the Kirchhoff‐Love model for pure bending of a thin solid symmetric plate under a transverse load.
Small‐amplitude homogenization is a procedure for computing the first correction in the homogenization limit. By using H‐measures, we calculate the first correction in the small‐amplitude homogenization limit of a non‐periodic sequence of tensors describing material properties in the Kirchhoff‐Love model for pure bending of a thin solid symmetric plate under a transverse load.
We consider two-phase multiple state optimal design problems for stationary diffusion equation. Both phases are taken to be isotropic, and the goal is to find the optimal distribution of materials ...within domain, with prescribed amounts, that minimizes a weighted sum of energies. In the case of one state equation, it is known that the proper relaxation of the problem via the homogenization theory is equivalent to a simpler relaxed problem, stated only in terms of the local proportion of given materials.
We prove an analogous result for multiple state problems if the number of states is less than the space dimension. In spherically symmetric case, the result holds for arbitrary number of states, and the optimality conditions of a simpler relaxation problem, which are necessary and sufficient, enable us to explicitly calculate the unique solution of proper relaxation for some examples. In contrary to maximization problems, these solutions are not classical.
