Shape Sensitivity Research Articles

Second‐order defeaturing estimator of manufacturing‐induced porosity on structural elasticity

AbstractManufactured metallic components often contain nonuniformly distributed pores of complex morphologies. Since such porosity defects have a significant influence on material behaviors and affect the usage in high‐performance applications, it is significant to understand the impact of porosity characteristics on the behaviors of components. In this work, a gradient‐enhanced porosity defeaturing estimator, which allows for the modeling of pore geometry and spatial distribution, is proposed within a general elastostatic framework. In this approach, the first‐order shape sensitivity is implemented to account for the change in the elastic quantity of interests to variations of pore sizes and shapes, which is then supplemented by a second‐order shape sensitivity whose mixed partial derivative quantifies the interactions between pores in proximity. The efficacy of the proposed method comes from its posterior manner that it only relies on field solutions of reference models where pores are suppressed. In this context, meshing difficulty and solution convergence issues are avoided, which would otherwise arise in a direct finite element analysis on porous structures. The impact of porosity on structural elastic performance is approximated using a second‐order Taylor expansion where the topological difference between the porous and reference domains is estimated by topological sensitivity; the field variables on pore boundaries are approximated as explicit functions of design variables using exterior formulations. Numerical results show that the elastic performances of components are influenced by the existence of pores. The pore‐to‐pore interactions are significant when pores are close by.

Shape Sensitivity of Eigenvalue Functionals for Scalar Problems: Computing the Semi-derivative of a Minimum

This paper is devoted to the computation of certain directional semi-derivatives of eigenvalue functionals of self-adjoint elliptic operators involving a variety of boundary conditions. A uniform treatment of these problems is possible by considering them as a problem of calculating the semi-derivative of a minimum with respect to a parameter. The applicability of this approach, which can be traced back to the works of Danskin [8, 9] and Zolésio [28], to the treatment of eigenvalue problems (where the full shape derivative may not exist, due to multiplicity issues), has been illustrated by Zolésio in [29] (see also [10, Chap. 10] and included references). Despite this, some of the recent literature (see, for example, [1] or [7]) on the shape sensitivity of eigenvalue problems still continue to employ methods such as the material derivative method or Lagrangian methods which seem less adapted to this class of problems. The Delfour–Zolésio approach does not seem to be fully exploited in the existing literature: we aim to recall the importance and the simplicity of the ideas from [8, 28], by applying it to the analysis of the shape sensitivity for eigenvalue functionals for a class of elliptic operators in the scalar setting (Laplacian or diffusion in heterogeneous media), thus recovering known results in the case of Dirichlet or Neumann boundary conditions and obtaining new results in the case of Steklov or Wentzell boundary conditions.

A level set-based topology optimization approach for thermally radiating structures

The need for efficient thermally radiating structures is apparent in many aerospace system designs including satellites, launch vehicles, and hypersonic aircraft. This paper presents a novel level set-based topology optimization approach for designing thermally efficient radiating structures. In this paper, we derive a shape sensitivity of the thermal heat power radiated objective function using the adjoint method. This sensitivity is a necessary ingredient for our gradient-based algorithm. We apply an augmented Lagrangian method to solve an example 2D problem where the goal is to maximize heat power rejected subject to a material volume constraint. The radiating surface is kept fixed during the optimization to maintain a design-independent boundary condition, while the conducting region is optimized. Several solutions are illustrated with varying initial conditions. We also present a case study indicating that maximizing the thermal compliance functional is not sufficient for solving this class of problems.

Fracture-based shape optimization built upon the topological derivative

In Silva et al. (2011) and Alidoost et al. (2020), the authors developed an approximation of the energy release rate field associated with a small edge or surface crack at any boundary location and with any orientation using the topological derivative. The approximation is computationally attractive because it requires only a single analysis on the non-cracked domain in contrast with conventional boundary-element and finite-element-based methods, which require a separate and costlier analysis for each crack length-location-orientation combination. In this work, a shape optimization scheme for fracture-resistant structures is developed using the energy release rate approximation. In the gradient-based optimization scheme, the domain and its boundary are defined implicitly using level-set functions. The level-set functions of arbitrary geometries are constructed using Boolean operations from the level-set functions of simple primitives. This geometrical representation has the dual advantage of (i) allowing shapes to intersect and/or separate during the optimization and (ii) simplifying the computation of the shape sensitivities.

Co-Based superalloy morphology evolution: A phase field study based on experimental thermodynamic and kinetic data

Cobalt-based superalloys with two phase γ/γ′ microstructures offer great promise as candidates for next-generation high-temperature alloys for applications, such as turbine blades. It is essential to understand the thermodynamic and kinetic factors that influence the microstructural evolution of these alloys in order to optimize the alloy compositions and processing steps with a goal to improve their coarsening, creep and rafting behavior. We are using a continuum phase field approach to study the diffusion process and to predict the equilibrium shapes of Co-Al-W γ′ precipitates. In order to obtain quantitatively predictive capabilities, we extract chemical free energies for the γ/γ′ phases based on CALculation of PHAse Diagrams (CALPHAD) thermodynamic data and diffusion mobilities for Co alloys based on CALPHAD kinetic data. We also use experimental or first-principles data for other quantities, such as misfit strain and interface information, for the parameterization of our model. A particular focus of our study is to understand how different energy balances, misfit strain and kinetics affect the coarsening and rafting behavior of γ′ precipitates, and the sensitivity of the final precipitate shape to materials parameters. We find that the equilibrium shape of the precipitate results from a delicate competition between chemical, interfacial, and elastic energies, and it is very sensitive to changes in model parameters. We examine how modeling input parameters affect the equilibrium shape of precipitates and relate these parameters to experimentally available values.

Performance Comparison of Quantized Control Synthesis Methods of Antenna Arrays

There is a great potential in small satellite technology for testing new sensors, processes, and technologies for space applications. Antennas need careful design when developing a small satellite to establish stable communication between the ground station and the satellite. This work is motivated by the design of an antenna array for a future rotatorless base station for the VZLUSAT group of Czech nano-satellites. The realized antenna array must cover a relatively broad range of elevation and azimuth angles, and the control must be fast enough to track the satellite in low Earth orbits. The paper deals with possibilities of synthesis of quantized control of the antenna array. It compares quantization influence for well-known deterministic synthesis methods. It shows the method for decreasing computational cost of synthesis using optimization approach and presents the multi-criteria optimization as a tool for reaching required radiation pattern shape and low sensitivity to quantization at the same time.

Velocity Field Level Set Method Incorporating Topological Derivatives for Topology Optimization

Abstract The velocity field level set method constructs the velocity field by velocity design variables and basis functions, and thus facilitates the use of general optimizers while still retaining the level set-based implicit topological representation. This paper incorporates the topological derivative concept into the velocity field level set method to enable automatic nucleation of interior holes. In each design iteration, a specified volume fraction of new holes is inserted at locations with smaller values of topological derivatives. Thus, the method provides a way to directly change the structural topology during the boundary evolution using the velocity field based on the shape sensitivity. Compared with the original velocity field level set method, the current implementation can further accelerate the topological and shape evolution during the optimization process. More importantly, the capability of hole nucleation eliminates the need of prescribing initial holes and thus alleviates the dependency of the optimized design on the initial design. Several numerical examples in both 2D and 3D design domains are presented to demonstrate the validity and efficiency of the proposed method.

Transient thermomechanical sensitivity analysis using a complex-variable finite element method

A complex-variable transient thermomechanical element was developed and used to compute highly accurate sensitivities of the thermal and stress time-dependent responses of a thick-walled cylinder with temperature-dependent material properties with respect to fifteen material, shape, and loading parameters using the complex Taylor series expansion method (CTSE). The analysis was conducted by employing a complex-variable user element within the Abaqus commercial finite element software, herein called ZFEM, using real and imaginary nodal degrees of freedom (DOF). The real coordinates denote the spatial coordinates, as typical, and the imaginary nodal DOF denote the perturbation of the geometry for shape sensitivities. Upon completion, the imaginary DOF hold the partial derivatives of the temperature and the displacements with respect to the parameter of interest. Sensitivities of stresses were obtained using the displacement sensitivities. The sensitivities were verified by a comparison with finite difference estimates using the built-in Abaqus elements. The numerical results demonstrate the superior accuracy, ease of use, and reduced run-time of ZFEM compared to the finite difference method. The conclusions show that ZFEM can be utilized to obtain time-dependent sensitivities with high accuracy and precision for transient thermomechanical analyses that are computationally more efficient in comparison to finite difference methods.

A level‐set‐based shape optimization method for thermoelectric materials

AbstractA thermoelectric device is a useful device which directly converts thermal energy to electric energy and vice versa. Although this device has such energy conversion functions, the demands in the industries are limited because of the relatively low conversion performance. In order to enhance this performance, the thermal conductivity of thermoelectric material should be low. According to dramatic improvements of nano‐processing accuracy in recent decades, many thermoelectric materials have been developed with low thermal conductivities by utilizing a unique phenomenon of a nanoscale heat conduction which is called “temperature jump.” However, few structural optimization methods have been proposed for nanoscale thermoelectric materials since a conventional analysis method cannot be applied to a heat conduction problem in a nanostructure. On behalf of the Poisson's equation, we need to introduce a Boltzmann transport equation for describing a nanoscale heat conduction problem with a discontinuity effect on a material interface. The level set method has a potential for dealing with such a boundary condition in structural optimization. That is, material interfaces can be expressed by the iso‐surface of the level set function, and the boundary effects are imposed in the boundaries during the optimization procedures. In this paper, we propose a level‐set‐based shape optimization method for thermoelectric materials. First, an analysis method for a nanoscale heat conduction problem based on the Boltzmann transport equation is introduced. An objective functional is defined for obtaining a low thermal conductivity material. Then, based on an adjoint method, we derive the shape sensitivity, which enables us to consider the effects of the temperature jumps on the material interfaces. Next, a level‐set‐based shape optimization is applied to our optimization problem. Finally, as a validation of our proposed method, several numerical examples are computed.

Damage Identification for Shear-Type Structures Using the Change of Generalized Shear Energy

Structural damage identification has become an important topic in the field of civil engineering in recent years. The shear-type structure, such as shear frame structure, is a common type used in civil engineering. In this paper, a damage identification method based on the change of generalized shear energy is proposed for shear-type structures. The main steps of the proposed method are as follows. Firstly, the element stiffness matrix in the structural finite element model is decomposed to obtain the elementary shear force vector. Secondly, the elementary generalized shear energy is calculated by the dot product of the vibration mode shape vector and the elementary shear force vector. Thirdly, structural damage locations can be determined by the changes of elementary generalized shear energy. Finally, more accurate damage localization and quantification are achieved by solving the mode shape sensitivity equation. A 20-storey numerical example and a three-storey experimental model are used to demonstrate the proposed damage identification algorithm. From the numerical and experimental results, it was found that the proposed approach can accurately identify the location and extent of the damage in the shear structures even if the data contain noise. It has been shown that the presented algorithm may be useful in the damage identification of shear-type structures.

Discrete adjoint methodology for general multiphysics problems

This article presents a methodology whereby adjoint solutions for partitioned multiphysics problems can be computed efficiently, in a way that is completely independent of the underlying physical sub-problems, the associated numerical solution methods, and the number and type of couplings between them. By applying the reverse mode of algorithmic differentiation to each discipline, and by using a specialized recording strategy, diagonal and cross terms can be evaluated individually, thereby allowing different solution methods for the generic coupled problem (for example block-Jacobi or block-Gauss-Seidel). Based on an implementation in the open-source multiphysics simulation and design software SU2, we demonstrate how the same algorithm can be applied for shape sensitivity analysis on a heat exchanger (conjugate heat transfer), a deforming wing (fluid–structure interaction), and a cooled turbine blade where both effects are simultaneously taken into account.

Sensitivity analysis and design optimization of 3T rotating thermoelastic structures using IGBEM

<abstract> <p>In this study, the isogeometric boundary element method (IGBEM) based on non-uniform rational basis spline (NURBS) is used to perform shape design sensitivity and optimization of rotating three-temperature (3T) thermoelastic structures. During the optimization process, the shape design sensitivity within the IGBEM formulation was derived to include precise geometries and greater continuities. It was found through the application of the IGBEM that the shape design velocity has a significant effect on accuracy of the obtained shape design sensitivity. As a result, the developed shape design sensitivity analysis (SDSA) technique based on the considered IGBEM formulation outperforms the computational solution based on the traditional SDSA method. The isogeometric shape sensitivity and optimal design for a complicated three-temperature thermoelastic problem in rotating structures are investigated. The impact of rotation on the thermal stress sensitivity, optimal three-temperature, optimal displacement and optimal three temperature thermal stress distributions are established. It is shown that the SDSA derived using IGBEM is efficient and applicable for most three-temperature thermoelastic optimization problems.</p> </abstract>

Concurrent shape optimization for macro- and micro-structure of 3D multiscale porous structure

In this study, we present a 3D shape optimization method for designing micro- and macro-structures concurrently. We assume the macrostructure consists of several arbitrary subdomains, which have different periodic microstructures. The macro- and microstructures are bridged by the homogenized elastic tensors, which are calculated by applying the homogenization method to the unit cells of the microstructures. Defining the boundary shapes of the macro-, microstructures and the interface shapes between the subdomains as design variable, the compliance of the macrostructure is minimized. The volume of the macrostructure considering the whole holes in the microstructures is used as the constraint. The homogenization equations for the microstructures and the equilibrium equation for the macrostructure are also used as the constraint. This design problem is formulated as a distributed-parameter optimization problem, and the shape sensitivity functions are theoretically derived. The optimum boundaries of the macro- and microstructures are determined by the H1 gradient method. The proposed method is applied to a numerical example to confirm the effectiveness of the proposed method.

Conformal topology optimization of multi-material ferromagnetic soft active structures using an extended level set method

Ferromagnetic soft active structures using embedded ferromagnetic particles in the soft polymer matrix can generate flexible locomotion and change configurations remotely, rapidly, and biologically friendly with an applied magnetic field. To achieve the desired motion, these soft active structures can be designed by tailoring the layouts of the ferromagnetic soft polymer. Although many magnetic soft active structures have been designed and fabricated, they are limited by the developer’s intuition and experience. Structural topology optimization has become a promising method to achieve innovative structures by optimizing the material layout, opening a new path for architecting ferromagnetic-driven active structures. Given the widespread adoption of thin-shell structures for soft robots, the extended level set method (X-LSM) and conformal geometry theory are employed to perform topology optimization of the ferromagnetic soft active structures on manifolds. The boundary evolution on a free-form 3D surface can be transferred into a 2D rectangular plane by solving a modified Hamilton–Jacobi equation weighted by conformal factors. The reconciled level set (RLS) method is firstly implemented within the X-LSM framework in this paper to enable the design of multi-material ferromagnetic soft active structures on free-from surfaces. The design objective consists of a subobjective function for kinematic requirement and a subobjective function for minimum compliance. The shape sensitivity was analyzed using the material time derivative and the adjoint variable approach. The proposed method was applied to design several single and multi-material ferromagnetic soft active structures. Two topologically optimized designs have been printed using functional 3D printing technology, or the so-called 4D printing, to physically realize soft active structures with built-in functionalities. The results of the numerical verification and experimental validation demonstrate the effectiveness of the proposed design and fabrication framework.

Isogeometric Shape Optimization for Design Dependent Loads

Abstract This paper presents a new isogeometric formulation for shape optimization of structures subjected to design dependent loads. This work considers two types of design dependent loads, namely, surface loads like pressure where the direction and/or magnitude of force changes with the variation of boundary shape, and body forces that depend on the material layout. These problems have been mostly solved by topology optimization methods which are prone to difficulties in determination of the loading surface for pressure loads and problems associated with non-monotonous behavior of compliance and low density regions for body forces. This work uses an isogeometric shape optimization approach where the geometry is defined using non-uniform rational B-spline and the control point coordinates and control weights of the boundary are chosen as design variables. This approach accommodates the design dependent loads easily, in addition to its other advantages like exact geometry representation, local control, fewer design variables, excellent shape sensitivity, efficient mesh refinement strategies, and smooth results that can be integrated with computer aided design. Two classes of optimization problems have been discussed; they are minimum compliance problems subjected to volume constraint and minimum weight problems subjected to local stress constraints. These problems are solved using convex optimization programs. Hence, expressions for full sensitivities are derived which are new for structural shape optimization problems with design dependent loads. Some representative engineering examples are solved and compared with existing literature to demonstrate the application of the proposed method.

Baśnie – wczoraj i dziś

Fairy tales play a key role in a child’s life because they stimulate emotional development, shape sensitivity, enrich vocabulary, and create social attitudes. Thanks to fairy tales, the child has the opportunity to identify with his favourite heroes, experience adventures, get used to difficult emotions and learn the rules governing the world. In addition, fairy tales have been passed down from generation to generation creating unbreakable bonds between the child and the parent, which affects their relationship and provides an attractive way of spending time together. By listening, reading and watching fairy tales, the child builds his worldview, shapes moral values, learns about positive models and prepares for a life in which he will have to make constant choices. Thanks to fairy tales, a child becomes more aware, interprets the various experiences of the characters, overcomes fear and becomes more flexible in overcoming the encountered difficulties. However, living in such dynamic times, with rapid technological development, a person should be prepared to receive modern fairy tales that differ in form, theme, personality of the protagonist, as well as extraordinary adventures of characters. Also, the adaptations of fairy tales produced in cinemas are famous for their excellent music, phenomenal special effects, and the selection of a talented cast. Therefore, while moving forward, man should not close himself to modernity. On the contrary, he should willingly participate in the reception of modern forms of fairy tales, which will make it easier for him to compare the classical form to modern one and provide knowledge, making it a sophisticated person, and also help encourage the child to reach for the book. In the article, the author describes the concept of a fairy tale and its necessary elements, emphasizes its significant impact on the development and education of a child. She also pays attention to the specificity of a modern fairy tale, detailing the differences between its traditional and contemporary form.

Application of the Complex Variable Step Method and the Boundary Element Method for Sensitivity Analysis of Steady Heat Conduction Problems

The paper concerns the problems related to applying the complex variable step method for the sensitivity analysis of the steady temperature field in the solid body domain due to the perturbations of the geometrical and physical parameters. The optimization problem using the approach proposed is also discussed. At the stage of numerical modelling, the boundary element method is used. The first part of the paper is devoted to the shape sensitivity. The results obtained are compared with the solution resulting from the implicit approach of sensitivity analysis. In the second part, the practical problem concerning optimizing the geometry of continuous casting mould cross-section is considered. The project variable vector contains the cooling pipes' radius and the volume flow rate of the cooling water. The numerical results and the conclusions are presented in the final part of the paper.

Multiscale modeling of fragmentation in an electrospray plume

We present a 3D-particle-in-cell (PIC) approach to modeling electrospray plumes typical of those formed by externally wetted emitter devices. Numerical grid-resolution techniques suitable for capturing strong electric fields in the emitter region were explored, and grid refinement criteria were quantified. The molecular dynamics simulations of the EMIM−BF4 ionic liquid system were modeled to determine the fragmentation mechanism in the presence of an electric field and dimer temperature as well as to provide fragmentation rates for the PIC simulations. An energy analysis of the molecular dynamics (MD) fragmentation demonstrated that the key mechanism for dimer fragmentation corresponds to a decrease in the Coulomb energy between the cation and anion in the system and that dimers of temperatures 300 and 600 K are extremely stable for electric fields smaller than 1.5 V/nm. Using probabilities of fragmentation consistent with the MD simulations, we implemented a dimer fragmentation model in our PIC simulations. The ion energy distribution functions obtained from the PIC simulations were used to predict retarding potential analysis (RPA) curves that were compared directly to measurements. The sensitivity of the RPA shape to the fragmentation probability was found to be significant. By comparing predicted and measured RPA curves for both negative and positive operation modes, and the fact that dimers do not fragment for electric fields less than 0.6 V/nm, we conclude that fragmentation of dimers occurs spontaneously due to their high thermal energies.

Influence of particle mineralogy and size on the morphological characteristics of mineral fillers

The morphological characteristics of mineral filler particles, such as contour shape, angularity and surface texture, are closely related to their mineralogy and size and the crushing process used. In the present study, the importance and effects of particle mineralogy and size on the morphology of different mineral fillers were examined. A total of 12 mineral filler fractions were prepared using the same grinding process and sieving method, and analyzed by digital imaging. Six parameters were used to describe the 2-dimensional characteristics of mineral filler particles based on digital image processing. The sensitivity of these parameters was analyzed by correlation and analysis of variance methods. The results show that for mineral fillers of the same particle mineralogy, as the particle size decreases, the contour shape becomes more regular and the angularity and surface texture becomes less abundant. For mineral fillers of the same particle size, the granite filler particles have a more regular contour shape and richer angularity and surface texture than the limestone and basalt filler particles. The form factor, angularity parameter AI1 and fractal dimension can best reflect the sensitivity of contour shape, angularity and surface texture characteristics of mineral filler particles to particle mineralogy and size.

A block forward substitution method for solving the hypercomplex finite element system of equations

The hypercomplex finite element method, ZFEM, allows the analyst to compute highly-accurate arbitrary-order shape, material property, and loading derivatives by augmenting the traditional finite element method with multiple imaginary degrees of freedom. In ZFEM, the real variables are converted to hypercomplex variables such as multicomplex, multidual, or quaternions. By uplifting the real variables to hypercomplex, derivatives are computed in an automated fashion using a standard finite element formulation. The use of multicomplex or multidual numbers provides higher-order derivatives. The drawback of ZFEM is that it increases the number of degrees of freedom of the real variable system by a factor 2n, where n is the order of the required derivative. In consequence, ZFEM increases the memory consumption and the solution time of the system of equations compared to the real variable system. The block forward substitution method (BFS), proposed in this work, addresses the memory and runtime issues. This new method solves the original real-valued FEM system once. Then, the derivatives are computed using pseudo-loads with the original system of equations. In contrast with the conventional solution method of ZFEM, BFS computes the hypercomplex contributions to the stiffness matrix element-wise, and it never assembles nor solves the full hypercomplex system of equations. In effect, the BFS method generalizes the first-order semi-analytical complex variable method to any order derivative. The BFS method provides the capability to allow a combination of real-variable and hypercomplex-variable elements within the same model. The numerical results indicate that a first-order derivative can be obtained with 1% to 8% additional computational time of the real-variable analysis. This allows the computation of multiple first-order derivatives by post-processing of a single FEM analysis. Additionally, it was shown that fourth-order shape sensitivities can be computed in less than 5% additional runtime of the real-variable FEM analysis.