Instytut Podstawowych Problemów Techniki

Streszczenie:

An elastic structure subjected to thermal and mechanical loading with prescribed external boundary and varying intern al interface is considered. The different thermal and mechanical nature of this interface is discussed, since the interface form and its properties affect strongly the structural response. The first-order sensitivities of an arbitrary thermal and mechanical behavioral functional with respect to shape and material properties of the interface are derived u sing the direct or adjoint approaches. Next the relevant optimality conditions are formulated. Some examples illustrate the applicability of proposed approach to control the structural response due to applied thermal and mechanical loads.

Słowa kluczowe:

multiphase structures, sensitivity analysis, optimization, heat transfer, thermo-elasticity

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The present paper provides the analysis of identification of damage or material parameters by using variation of modal characteristics (eigenvalues and eigenmodes), static or dynamic compliance and temperature fields. The distance norm between the measured and predicted by model data is assumed to depend also on an additional control parameter introduced in the system in order to increase sensitivity with respect to identified parameters. In the case of frame and plate structures such assisting control can be executed by additional supports, masses or loads introduced into a structure. In the case of thermal loading, the distribution of heat sources or boundary temperature may constitute control parameters. The proposed methodology is illustrated by analyzing identification of location and magnitude of damage in beams and plates by means of variation of free frequency spectrum and the related free vibration modes. Similarly for the thermal problem the heat flux distribution on the boundary is specified in order to increase the sensitivity of the assumed distance norm.

Słowa kluczowe:

Damage identification, Modal analysis, Static, dynamic and thermal loads, Assisting control parameters

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The detection of defects as well as their location, orientation and size is performed using measurements of surface temperature either at some selected points or on selected surface areas or lines. The response temperature of a structure is caused by statically, quasi-statically or dynamically applied thermal load on some structural boundary parts or within its domain. On the basis of results of measurements, an inverse heat transfer problem is formulated for a model structure and next solved. The inverse solution is constructed by minimizing the properly defined distance norm of measured and model temperatures. The model temperature distribution is calculated using the finite element model of a structure, while in minimizing the distance norm functional the gradient-oriented methods are used. The proper sensitivities of introduced identification functional are also derived. Some simple examples illustrate the applicability of the proposed approach.

Słowa kluczowe:

identification, sensitivity, thermographic methods, path-independent integrals

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Streszczenie:

A heat conduction problem within a thermal anisotropic solid body is formulated. With respect to an arbitrary thermal functional defined over space domain, its first-order sensitivities with respect to variation of structural material parameters as well as external boundary and internal interfaces are derived using the direct and adjoint approaches. Next, the application of derived sensitivity expressions in optimizing thermal systems is presented and illustrated by some examples.

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Large structures are usually composed of elements by properly designed connections. The optimal design solution in such cases should provide optimal size and number of elements together with optimal connection stiffness. The problem is formulated by assuming the element cost to be a nonlinear function of its size and the cost of connection to depend on its stiffness or transmitted forces. The number of elements and the connection stiffness now constitute the design parameters to be determined. A two-level procedure is proposed for determination of the optimal segmentation for beam and plate structures. Several illustrative examples are presented.

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Streszczenie:

Considering an arbitrary behavioural functional specified for a mixed Dirichlet-Neumann boundary-value problem, its first variation with respect to variation of domain shape is derived using the material derivative concept and adjoint approach. Next the invariance of this functional is proved for the case of translation, rotation and scale change of an arbitrary homogeneous and isotropic domain of a problem. The associated path-independent integrals are derived and it is shown that the sensitivity analysis with respect to translation, rotation or expansion of defects can be performed by using these integrals

Słowa kluczowe:

Numerical method, Boundary-value problems, Path integral, Sensitivity analysis, Holes, Optimization, Geometrical shape

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Streszczenie:

The semi-analytical, analytical and direct methods for numerical structural shape sensitivity analysis are discussed for a beam model and the general three-dimensional case. While the two first methods are applied directly to the finite element model of a structure, the direct approach follows from a continuous formulation and only the final results can be discretized.

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Disks, plates or beams subjected to loading and initial strains or displacements are assumed to be composed of portions interconnected by hinge or slip and dilatancy lines admitting discontinuities in displacements or slopes. The shape of such lines and their stiffness properties are subjected to variation. The sensitivity analysis is first discussed for an arbitrary functional of generalized stress, strain, displacement and boundary traction. The variation of complementary and potential energies is considered as a particular case of a general derivation for sensitivity. The optimal design problem is then considered and the relevant optimally conditions are derived. The general theory is illustrated by examples of sensitivity analysis and optimal design with respect to shape, position and stiffness of discontinuity lines for disks, plates and beams. Numerical aspects of sensitivity analysis are also discussed.

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Disks and plates subjected to stretching and flexure are stiffened by curvilinear ribstiffeners whose shape and cross-sectional area are subject to optimization. Sensitivity analysis is first discussed for an arbitrary integral functional of generalized stress, strain and displacement and boundary domains. The optimality conditions are next derived. The general theory is illustrated by three examples of optimal design of ring-stiffeners in circular disks and plates and of rib-stiffeners in a rectangular plate.

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The method of calculating the sensitivities of buckling loads and vibration frequencies of nonlinear plates with respect to shape functions is presented. These shape functions describe the external shape of an unstiffened plate or the shape of a stiffener within the domain of a plate with fixed external boundaries. An adjoint variable method is used to determine the sensitivities. Thus, only one additional solution for an adjoint plate is required.

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A plane arch with variable form or cross-sectional stiffness is considered. The associated variation of any functional that depends on generalized stress, strain, and displacement is explicitly expressed, using primary and adjoint fields. Geometrical linear and material nonlinear behavior of an arch are considered. The optimal design problem is discussed and relevant optimality conditions are derived. Illustrative examples are presented. The analysis presented can be used in optimization of curvilinear stiffeners on disks and plates.

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Considering arbitrary stress, strain or displacement functionals specified over a domain of an elastic, homogeneous and isotropic body, their invariance is proved for the case of translation, rotation and scale change of an arbitrary domain within the body. The assosciated class of path-independent integrals is derived. It is shown that sensitivity analysis with respect to translation, rotation or expansion of defects can be performed by using these path-independent integrals.

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For a linear elastic structure, the first variations of arbitrary stress, strain and displacement functionals were explicitly expressed in terms of variations of material stiffness parameters within specified domain, in References 1 and 2, by using the solutions for primary and adjoint systems. In this paper, explicit expressions for second variations are derived for the case when the material stiffness or compliance variations depend on a set of parameters. These expressions arc obtained in terms of variations of stiffness or compliance parameters by using the solutions of two additional sets of adjoint systems. Furthermore, the variation of local stress, strain or displacement is considered. Several illustrative examples are provided for the first- and second-order sensitivity analysis of beams.

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The incremental problem for a brittle-plastic body is considered for which stable propagation of a damage interface occurs within the material. The stability conditions are derived and applied to several simple illustrative examples.

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For a linear elastic structure, the first variation of an arbitrary stress, strain and displacement functionals corresponding to variation of shape of external boundaries or interfaces is derived by using the solutions for primary and adjoint systems. The application to optimal design is next presented and the relevant optimally conditions are derived from general expressions. The path-independent integrals used in fracture mechanics are rederived as a particular case of general expressions.

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For a linear elastic structure, the first variation of an arbitrary stress, strain and displacement functionals corresponding to variation of material parameters within specified domain is derived by using the solution for primary and adjoint systems. This variation is of fundamental importance in sensitivity analysis, optimal design and identification problems. Simple examples of optimal stiffness design and identification of stiffness parameters in beams are presented.

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Streszczenie:

The problem of optimal design of the shape of a free or internal boundary of a body is formulated by assuming the boundary shape is described by a set of prescribed shape functions and a set of shape parameters. The optimzation procedure is reduced to determination of these paramenters. For constant volume or material cost constraint, the optimality conditions are derived for the case of mean complicance design of elastic structures of a non-linear material. Some additional conditions for the global minimum of the mean compliance are proved. The most typical cases of boundary variations are discussed. The optimal shape problem is next formulated by means of the finite element method and the iterative solution algorithm is discussed by using the optimalitycriteria. Several simple numerical examples are included.

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