In this paper, we investigate the finite element discretization of the isotropic Helmholtz-type PDE filter for topology optimization and propose a robust alternative based on mass lumping. While the new strategy preserves the filter’s efficiency, it enforces the discrete maximum principle, ensuring that the filtered design variables remain strictly within the admissible interval. We establish theoretical support for the approach, and comprehensive numerical experiments validate its reliability and effectiveness. Numerical results show that the proposed approach provides virtually identical results to those of the standard PDE filter for radii larger than the characteristic mesh size. It also delivers improved accuracy and stability when the radius is comparable to or smaller than the characteristic mesh size. In topology optimization, this may lead to distinct designs and provide enhanced numerical stability within this range.

This study proposes an optimization methodology to find optimal heat source paths for point-melting in Electron Beam Melting (EBM) Powder Bed Fusion (PBF) processes, aiming to reduce the need for support structures and improve print quality. The building process is simulated using a time-dependent, one-way coupled, non-linear thermo-mechanical model, assuming negligible molten flow, with elastoplastic behavior and temperature-dependent material parameters. The goal of the optimization problem is to find heat source paths that minimize a global temperature measure with a penalty on excessive local temperatures. The numerical methodology is based on solving the non-linear partial differential equations via the Finite Element Method (FEM) and is applied in numerical examples for printing with titanium alloy Ti6Al4V. Metrics related to heat, residual displacement, and residual stresses are considered to assess the performance of different point-melting strategies and to compare optimized and conventional paths. The feasibility of the proposed optimization methodology for practical applications and alternatives towards future methodological advancements are discussed. The study provides a Python-based, MPI-parallelized implementation using open-source libraries and is made available for further research and applications.

This study presents a Topology Optimization (TO) methodology for determining the optimal distribution of support structures in the powder bed fusion Additive Manufacturing (AM) fabrication technology. The support structures enhance thermal dissipation, reducing distortion on overhang surfaces during the build process and improving printability. A novel transient thermoelastic layer-by-layer model simulates the AM process, evaluating temperature and displacement in partially built structures with supports. We propose two new objective functions related to part distortion, perform their adjoint sensitivity analysis, and present a comprehensive set of numerical experiments with different geometries and levels of overhang complexity. The numerical results show non-standard optimal support structures obtained using the proposed TO-AM methodology for different model parameters, including support volume, thermal anisotropy, time discretization method, and building times for each layer.

Among the wide range of structural polymers currently available, this work deals with high-density polyethylene (HDPE). The typical viscoelastic behavior of this material is not trivial to model and has already been investigated by many authors. We employ the fractional Zener model to fit our experimental creep results of HDPE evaluated at different stress levels. This model produces fractional constitutive equations with excellent curve-fitting properties and fewer parameters to be identified in relation to traditional models. The results are compared with those ones provided by the application of the Prony series method. The first novelty of this paper is the application of the time-stress equivalence principle (TSEP), coupled to the fractional model, to estimate creep at intermediate stress levels, that in turn, were not measured experimentally but lie within the stress range used to calibrate the model. We compare the results provided by this method with those based on linear interpolation of the parameters. Although there is clear benefits requiring fewer parameters, fractional derivatives render costly computations due to their history memory. To cope with this, we propose a new algorithm, called GPE, which shows a compromise between enhanced efficiency and accuracy when compared with other proposals of the literature. These features are verified with simulations for simple functions, and a long term creep test with the fractional Zener model. The combined application of fractional derivatives, TSEP and the new GPE algorithm results in a novel efficient and effective alternative to account for the creep modeling of HDPE.

A Correction to this paper has been published: 10.1007/s00466-022-02145-2

A topology optimization formulation including a model of the layer-by-layer additive manufacturing (AM) process is considered. Defined as a multi-objective minimization problem, the formulation accounts for the performance and cost of both the final and partially manufactured designs and allows for considering AM-related issues such as overhang and residual stresses in the optimization. The formulation is exemplified by stiffness optimization in which the overhang is limited by adding mechanical or thermal compliance as a measure of the cost of partially manufactured designs. Convergence of the model as the approximate layer-by-layer model is refined is shown theoretically, and an extensive numerical study indicates that this convergence can be fast, thus making it a computationally viable approach useful for including AM-related issues into topology optimization. The examples also show that drips and sharp corners associated with some geometry-based formulations for overhang limitation can be avoided. The codes used in this article are written in Python using only open sources libraries and are available for reference.

This paper presents a general thermodynamically consistent non-isothermal phase field framework to model effects of damage, fracture and fatigue evolutions in elasto-plastic materials under the hypothesis of small strains. The proposed methodology is obtained from the principle of virtual power, energy balance and second law of thermodynamics in the form of a generalized Clausius–Duhem inequality for entropy. Aspects of the energy degradation functions are thoroughly investigated for an isotropic elasto-plastic material with viscous dissipation and constant specific heat. A new combined degradation function is proposed to degrade both elastic and plastic energy densities as an alternative to the classical degradation functions. Our conceptual framework leads to thermodynamically consistent models that may include contributions not usually considered in the literature, as temperature and inertia effects as well as time-rate dependent processes. The governing nonlinear transient equations obtained are solved adopting a semi-implicit time integration scheme combined with the classical Newton–Raphson iterative procedure. Results for an I-shaped specimen made of 7075-T7351 aluminum alloy under different conditions are presented. The proposed model is able to reproduce qualitative and quantitatively the ductile fracture and the fatigue phenomena. In particular for fatigue, both the S–N experimental data and the Paris crack growth curve over cycles are recovered. In addition, the cycle jump strategy reduces four times the CPU time for fatigue simulations.

In this work, we apply semi and fully-implicit time integration schemes to the damage and fatigue phase field presented in Boldrini et al. (2016). The damage phase field is considered a continuous dynamic variable whose evolution equation is obtained by the principle of virtual power. The fatigue phase field is a continuous internal variable whose evolution equation is considered as a constitutive relation to be determined in a thermodynamically consistent way. In the semi-implicit scheme, each equation is solved separately by suited implicit method. The Newton's method is used to linearize the equations in the fully-implicit scheme. The time integration methods are compared and the results of damage and fracture evolution under the influence of fatigue effects are presented. The computational cost associated to the semi-implicit scheme showed be lower than the fully counterpart.

Considerations on the derivatives of Gâteaux and Fréchet ResumoEm matemática, é interessante investigar as conexões que existem entre temas abordados em diferentes contextos.Em relac ¸ão à derivada de Fréchet surge a questão: que conexões podem ser estabelecidas entre a derivada de Fréchet e o cálculo variacional?Por meio de pesquisa bibliográfica, expomos neste trabalho a definic ¸ão de derivada direcional e da primeira variac ¸ão segundo Lagrange.Apresentamos também algumas considerac ¸ões sobre as derivadas de Gâteaux e Fréchet e introduzimos a diferenciabilidade estrita e sua relac ¸ão com a derivada de Fréchet.Para mostrar as conexões existentes entre as derivadas apresentadas, exploramos alguns exemplos de cálculo de derivadas de Gâteaux e Fréchet e como uma aplicac ¸ão expomos a soluc ¸ão do problema de mínimos quadrados e calculamos a derivada de Fréchet de dois funcionais integrais, um oriundo do método variacional aplicado a um problema de contorno linear e outro de um funcional que surge em problemas de cálculo variacional.

Phase field equations are used to model a wide range of multiphase problems such as separation of fluids, solidification, viscous fingering, fracture and fatigue. A wide variety of methods to numerically solve phase field equations can be found in the literature. In particular, high order methods are an effective option when accuracy improvement is desired. In the first part of this work, we analyze the accuracy and computational efficiency of the high order finite element method (FEM) and discontinuous Galerkin (DG) method applied to the second-order Allen-Cahn (AC) and fourth-order Cahn-Hilliard (CH) equations. Several schemes for time integration are used for these equations. The explicit schemes are the forward Euler, classical fourth-order Runge-Kutta (RK4) and the strong stability preserving ten stages fourth-order Runge-Kutta (RKSSP-10,4) described in Gottlieb et al. (2011). The backward Euler and trapezoidal implicit methods are adopted in the full and semi implicit schemes, as proposed in Eyre (unpublished). Manufactured solutions for one dimensional problems are used in order to evaluate the errors and to compare the different numerical methods. By choosing an adequate discretization for AC equations resulting from the previous analysis of the first part of the work, in the second part, we propose a numerical semi implicit scheme to solve the damage and fracture model described in Boldrini et al. (2016). This procedure employs the FEM for spatial discretization, the Newmark method for time integration of the kinematics equation and the backward Euler for the damage phase field evolution. Finally, results for 2D benchmark tests are presented for the fracture phase field model and the convergence to a sharp crack for a small width of the damage phase field layer gamma is verified. (C) 2017 Elsevier Ltd. All rights reserved.

The evaluation of structural response derivatives with respect to design parameters, usually known as sensitivity analysis, is an issue of paramount importance in gradient-based optimization and reliability analyses in engineering. In the last 20 years, much research has been devoted to develop efficient strategies for the accurate evaluation of sensitivity information. A relatively new and promising procedure combines the semianalytical (SA) approach with the use of complex variables (CVSA). This method allows the use of diminutive perturbations, circumventing the weakness that the traditional SA approach shows when applied to shape design variables. In spite of the great potential of the CVSA, its formulation and application has been restricted to path independent problems. In this chapter we aim to extend the method to handle path dependent problems, emphasizing the treatment of internal variables, such as accumulated plastic strain and damage. In order to make the concept easy to understand, we use the method to evaluate the sensitivity of particular homogenized properties of a 2D periodic truss material (PTM). Optimization of PTMs has encountered great potential in tissue engineering, as well as in automotive and aeronautical applications. Generally PTMs are designed to operate in the linear geometrical and constitutive range. However, using sensitivity analysis we can obtain an insight about how these designed homogenized properties behave when geometrical and/or material nonlinearities are considered.

The application of gradient-based methods to structural optimization problems usually requires the determination of displacement sensitivities with respect to design variables. In this regard, it is important to have at hand comprehensive methods for sensitivity analysis, which show stability, efficiency and accuracy. Particularly, application of the semi-analytical method for linear and non-linear problems is generally a good trade-off between formulation simplicity and accuracy. In spite of that, semi-analytical methods are known to behave pathologically for shape design variables when the structure is subjected to rigid rotations. A large number of solutions for this problem have been presented in last years, although the formulation involved is generally not trivial, especially in the nonlinear case. A recent method, which adopts the semi-analytical approach and uses complex variables has rendered very promising results for all the aforementioned aspects: stability, efficiency and accuracy. Additionally, it is simple to codify. The present contribution is concerned with the application of this sensitivity analysis method to geometrically nonlinear truss problems. To this end, a finite element formulation is presented and displacement sensitivities are evaluated with respect to material and shape design variables. The results are compared to those obtained using the semi-analytical method with real variables and to global finite differences. An example demonstrates the potentiality of this new approach.

Neste trabalho realiza-se um abrangente estudo que visa a aplicacao do metodo semi-analitico de analise de sensibilidade, utilizando variaveis complexas (SAC) em estruturas trelicadas, considerando o comportamento nao linear geometrico e material. Tal pesquisa foca principalmente nos problemas dependentes da trajetoria e no tratamento adequado para a atualizacao das variaveis internas, sendo este um aspecto nao encontrado na revisao bibliografica pelo autor e aplicavel em problemas que envolvem plasticidade e dano. Estudos anteriores mostram que em problemas independentes da trajetoria o metodo SAC apresenta grande eficiencia e economia de armazenamento, uma vez que as operacoes sao realizadas no nivel do elemento. Verifica-se que este metodo quando aplicado em problemas dependentes da trajetoria, apresenta amesma eficiencia detectada na contraparte independente comumcusto de armazenamento um pouco mais elevado. Contudo, as operacoes ainda se mantem no nivel do elemento. Com a finalidade de realizar tal estudo, a formulacao de elementos finitos e as diferentes metodologias para avaliar a sensibilidade de respostas estruturais, tanto para problemas dependentes quanto independentes da trajetoria, sao apresentadas em detalhes. Por fim, realizasse um estudo comparativo entre os diferentes metodos de sensibilidade em problemas que sejam dominados por rotacao de corpo rigido, que possuam descontinuidades nos coeficientes da sensibilidade e em estruturas celulares.