Instytut Podstawowych Problemów Techniki

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This paper addresses a challenging engineering problem that combines stress-limited topology optimization, reliability analysis, and plasticity-based low-cycle fatigue. Each of these issues represents a complex problem on its own, necessitating significant computational effort. In this study, we propose a novel approach that integrates safety assessment into the topology optimization process while considering the number of cycles for low-cycle fatigue. Our method employs a linear approximation of the performance function for safety control, incorporating the number of failure cycles within a complex, multi-level load program. The methodology is validated through real experiments, using a finite element model with cubic shape functions that yield nearly identical results between numerical and experimental outcomes in the case of fatigue-resistant design for a bi-axially tensioned structural joint.

Słowa kluczowe:

Topology optimization, stress constraints, Reliability analysis, low-cycle fatigue, fatigueplasticity

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Streszczenie:

This study presents a general computational framework for topology optimization under constraints related to various engineering design problems, including: reliability analysis, low-cycle fatigue assessment, and stress limited analysis. Such a framework aims to facilitate comprehensive engineering design considerations by incorporating traditional constraints such as displacement and stress alongside probabilistic assessments of fatigue failure and the complex behaviors exhibited by structures made of elastoplastic material. The framework's amalgamation of diverse analytical components offers engineers a versatile toolkit to address intricate design challenges. Notably, the inclusion of reliability analysis introduces a probabilistic perspective, transforming conventional design constraints into random parameters, thereby enhancing the robustness of the design process.

Key to the framework's efficacy is its implementation using MATLAB mathematical computing software, leveraging the platform's efficient code execution and object-oriented programming paradigm. This choice ensures an intuitive and potent environment for designers and researchers, facilitating seamless adaptation across various engineering applications. Additionally, the proposed previously by the Authors algorithm for the topology optimization is extended by adaptive strategy allowing for efficient adjustment of an amount of material removed at individual optimization step.

The presented framework is offering a comprehensive and integrated approach to address multifaceted design challenges while enhancing design robustness and efficiency.

Słowa kluczowe:

Topology optimization, Stress constraints, First order reliability analysis, Low-cycle fatigue, Plasticity

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The objective of this study is to propose a relatively simple and efficient method for reliability based topology optimization for structures made of elasto-plastic material. The process of determining the optimal topology of elasto-perfectly plastic structures is associated with the removal of material from the structure. Such a process leads to weakening of structural strength and stiffness causing at the same time increase the likelihood of structural failure. An important aspect of engineering design is to track this probability during the optimization process and not allow the structure safety to exceed a certain level specified by the designer. The purpose of this work is to combine the previously developed yield-limited topology optimization method with reliability analysis using first order approach. Effectiveness of the proposed methodology is demonstrated on benchmark problems proposed by Rozvany and Maute, and the elasto-plastic topology design of L-shape structure which is frequently used in different approaches for stress constrained topology optimization.

Słowa kluczowe:

topology optimization, reliability analysis, elasto-plastic analysis

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Streszczenie:

This study is devoted to a novel method for topology optimization of elastoplastic structures subjected to stress constraints. It should be noted that in spite of the classical solutions of the different type of elastoplastic topology problems are more than 70 years old, the integration of the Prandtl-Reuss constitutive equations into the topology optimization process is not very often investigated in the last three decades. In the presented methodology where the classical variational principles of plasticity and the functor-oriented programming technique are applied in topology design, the aim is to find a minimum weight structure which is able to carry a given load, fulfills the allowable stress limit, and is made of a linearly elastic, perfectly plastic material. The optimal structure is found in an iterative way using only a stress intensity distribution and a return mapping algorithm. The method determines representative stresses at every Gaussian point, averages them inside every finite element using the von Mises yield criterion, and removes material proportionally to the stress intensities in individual finite elements. The procedure is repeated until the limit load capacity is exceeded under a given loading. The effectiveness of the methodology is illustrated with three numerical examples. Additionally, different topologies are presented for a purely elastic and an elastoplastic material, respectively. It is also demonstrated that the proposed method is able to find the optimal elastoplastic topology for a problem with a computational mesh of the order of tens of thousands of finite elements.

Słowa kluczowe:

topology optimization,elastoplastic structures,minimum-weight design,stress constraints

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Streszczenie:

The paper deals with a novel approach to development of optimality criteria based finite element code for topology optimization of elasto-plastic structures. The novelty of this work is related to the concept of function object called functor and its application to efficient FE code development. First, the general problem of topology optimization under stress constraints is briefly formulated. Then, the programming aspects of topology optimization using traditional object-oriented and functor based programming are discussed. The advantages of the functor based approach are related to simplicity of designing the FE code architecture and reusability of this code. In particular the metric known as 'Lack of cohesion of methods' is useful in comparing these two different paradigms. Finally, the paper is also illustrated with numerical examples of topology optimization using the proposed methodology.

Słowa kluczowe:

topology optimization, function object, functor programming, optimal design, elasto-plastic structures, finite element programming

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The purpose of this study is to present an optimal design procedure for elasto-plastic structures subjected to impact loading. The proposed method is based on mode approximation of the displacement field and assumption of constant acceleration of impacted structure during whole time of deformation process until the plastic displacement limit is reached. Derivation of the method begins with the application of the principle of conservation of linear momentum, followed by determination of inertial forces. The final stage of the method utilizes an optimization technique in order to find a minimum weight structure. Eventually, effectiveness and usefulness of the proposed method is demonstrated on the example of a planar truss structure subjected to dynamic loading caused by a mass impacting the structure with a given initial velocity.

Słowa kluczowe:

structural dynamics, optimal design, elasto-plastic structures, short-time dynamic loading

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The aim of this paper is to create new type of plastic limit design procedures where the influence of the limited load carrying capacity of the beam-to-column connections of elasto-plastic steel (or composite) frames under multi-parameter static loading and probabilistically given conditions are taken into consideration. In addition to the plastic limit design to control the plastic behaviour of the structure, bound on the complementary strain energy of the residual forces is also applied. If the design uncertainties (manufacturing, strength, geometrical) are taken into consideration at the computation of the complementary strain energy of the residual forces the reliability based extended plastic limit design problems can be formed. Two numerical procedures are elaborated. The formulations of the problems yield to nonlinear mathematical programming which are solved by the use of sequential quadratic algorithm.

Słowa kluczowe:

reliability analysis, limit analysis, residual strain energy, Monte Carlo simulation, optimal design

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Streszczenie:

The aim of this paper is to take into consideration the influence of the limited load carrying capacity of the connections on the plastic limit state of elasto-plastic steel (or composite) framed structures under multi-parameter stochastic loading and probabilistically given conditions. In addition to the plastic limit design to control the plastic behaviour of the structure, bound on the complementary strain energy of the residual forces is also applied. This bound has significant effect for the load parameter. If the design uncertainties (manufacturing, strength, geometrical) are expressed by the calculation of the complementary strain energy of the residual forces a reliability based extended limit design problem is formed. The formulations of the problems yield to nonlinear mathematical programming which are solved by the use of sequential quadratic algorithm. The bi-level optimization procedure governed by the reliability index calculation.

Słowa kluczowe:

limit analysis of frames, reliability analysis, optimization

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Streszczenie:

Optimal topologies obtained for structures subjected to deterministic loading can be sensitive to loading variations in terms of both magnitude and direction. Therefore, in this study we consider problem of topology optimization for structures subjected to probabilistic loading. The proposed method applies basic findings from probability theory, which allow to transform the original problem of topology optimization under single probabilistic loading into analogous problem of topology optimization under multiple deterministic loading cases. After recalling the theoretical background of the method,' its effectiveness is demonstrated on an examples of cantilever structure subjected to horizontally oriented load with randomly varying angle of action.

Słowa kluczowe:

Topology optimization, Stochastic load, Elastoplastic FE analysis

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Streszczenie:

The aim of the paper is a practical engineering formulation of the topology optimization for
structures made of elastoplastic material. This paper provides a comprehensive approach to
optimizing the topology of elastoplastic structures, including both the problem statement and its
efficient computer implementation. Instead of the traditional compliance minimization approach,
the aim of this work is to find a structure with the minimum mass that can carry on a given load,
provided that the corresponding stresses do not exceed an acceptable limit. The general formulation
of the problem is based on the classical approach allowing to determine the yield strength of the
designed structure [4]. The proposed method finds the optimal structure in an iterative way, using
only the stress intensity distribution and does not require the user to explicitly know sensitivity.

Słowa kluczowe:

topology optimization, stress constraints, optimal design

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Streszczenie:

To solve the problem of reliability-based topological optimization, a heuristic algorithm was used, consisting in removing redundant material in the areas with the lowest stress intensity. In this algorithm, the design variables represent the material densities in the individual finite elements. The material is removed by reducing the density of finite elements as a function of the stress intensity.

Słowa kluczowe:

topology optimization, reliability analysis, first order approximation

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Streszczenie:

Plasticity in topology optimization is almost forgotten direction in this popular research area. This paper presents a recently
developed elasto-plastic topology optimization procedure extended with reliability constraint. It recalls fundamental concepts from
reliability analysis and introduces an algorithm for topology optimization of elasto-plastic structures. In addition to the elasto-plastic
constitutive law of the applied material the optimization algorithm includes stress constraints, as well. The presented numerical examples
show dependence of the volume fraction on probability of failure.

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Streszczenie:

This study is devoted to a practical method for topology optimization of elastoplastic structures subjected to stress constraints. Instead of the classical compliance minimization problem the aim of this work is to find a minimum weight structure, which is able to carry given load and the corresponding stresses do not exceed an allowable limit. The general form of the problem is based on the classical limit design formulations of plasticity. The proposed method finds optimal structure in an iterative way using only stress intensity distribution and doesn't require computing of any gradients or sensitivities. Our method starts with determining representative stresses in every quadrilateral finite element. At first an elastoplastic analysis is performed to obtain stress values in four Gaussian points, then by the use of von Misses criterion and these stress values the resultant stress is calculated. Next, having obtained stress intensity distribution within the structure we apply penalization to avoid stress concentration issues. Finally, the material is removed proportionally to the stress intensities of individual finite elements. The above mentioned procedure is repeated until limit load capacity is achieved for a given loading vector. The checkerboard problem is solved by means of design filter. Two benchmark problems have been selected as illustrative examples. They are: cantilever and simply supported beam. For these examples parametric studies on different length to height ratios and support patterns are conducted. Additionally, the results of topology optimization for different values of filter radius and penalty parameter are presented.
Finally, efficient computer implementation based on functor-oriented programming is discussed. It is demonstrated how Functor and Template-based programming can be utilized to create versatile Finite Element environment. Within this environment computation of all element matrices and loading vectors can be called in the same way, this in turn allows for implementation of effective aggregation procedure.

Słowa kluczowe:

topology optimization, minimum-weight design, functor-oriented programming, stress constraints

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Streszczenie:

The subject of this study is an efficient approach to the development of a finite element framework, which is intended to be used for solving a variety of problems in computational solid mechanics. One of such problems, recently becoming an active field of research, is topology optimization of structures made of elastic-plastic materials. For finding the optimal topology of real, practical and complex structures the knowledge of a number of numerical algorithms is required, to mention a few: modification of finite element meshes, aggregation of tangent stiffness matrices, or direct and iterative solvers. The classical computer implementation of the original Classical Optimality Criteria method (COC) of the topology optimization problem given by Bendsoe and Sigmund is relatively simple and contains 99 lines of code in the MATLAB language. However, it assumes that there exists only a single loading case, single displacement (compliance) constraint, the material is linearly elastic and the optimal topology can be found using the so-called Solid Isotropic Material with Penalization (SIMP) algorithm, which is based on the original COC method. In reality, engineers face a slightly different problem. They need to find the topology of a minimum weight structure subjected to multiple loading cases, made of an elasto-plastic material, and with a limit on stresses. The above mentioned SIMP approach may not lead to an optimal solution.

Słowa kluczowe:

functor-oriented programming, topology optimization, elastoplastic FE analysis

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