Shape Gradient Research Articles

A discontinuous Galerkin level set method using distributed shape gradient and topological derivatives for multi-material structural topology optimization

A discontinuous Galerkin level set method using distributed shape gradient and topological derivatives for multi-material structural topology optimization

Lagrangian approach and shape gradient for inverse problem of breaking line identification in solid: contact with adhesion

A class of inverse identification problems constrained by variational inequalities is studied with respect to its shape differentiability. The specific problem appearing in failure analysis describes elastic bodies with a breaking line subject to simplified adhesive contact conditions between its faces. Based on the Lagrange multiplier approach and smooth Lavrentiev penalization, a semi-analytic formula for the shape gradient of the Lagrangian linearized on the solution is proved, which contains both primal and adjoint states. It is used for the descent direction in a gradient algorithm for identification of an optimal shape of the breaking line from boundary measurements. The theoretical result is supported by numerical simulation tests of destructive testing in 2D configuration with and without contact.

Individual differences in stimulus identification, rule induction, and generalization of learning.

In the field of stimulus generalization, an old yet unresolved discussion pertains to what extent stimulus misidentifications contribute to the pattern of conditioned responding. In this article, we perform cluster analysis on six datasets (four published datasets and two unpublished datasets, included N = 950) to examine the relationship between interindividual differences in (a) stimulus identification, (b) patterns of generalized responding, and (c) verbalized generalization rules. The datasets were obtained from online predictive learning tasks where participants learned associations between colored cues and the presence or absence of a hypothetical outcome. In these datasets, stimulus identification and expectancy ratings were assessed in separate phases to a range of colors varying between blue-green. Using cluster analyses on performance during stimulus identification, we identified different subgroups of participants (good vs. bad identifiers). In all six datasets, we found a close relationship between the pattern of stimulus identification and the shape of the expectancy gradient across the test dimension between the identified subgroups. Furthermore, participants classified as good identifiers were more likely to report a similarity generalization rule than a relational or linear rule, suggesting that individual differences in stimulus identification are related to individual differences in generalization rules. These findings suggest that greater consideration should be given to interindividual variability in stimulus identification, inductive rules, and their relationship in explaining patterns of generalized responses. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Early radial positional information in the cochlea is optimized by a precise linear BMP gradient and enhanced by SOX2

Positional information encoded in signaling molecules is essential for early patterning in the prosensory domain of the developing cochlea. The sensory epithelium, the organ of Corti, contains an exquisite repeating pattern of hair cells and supporting cells. This requires precision in the morphogen signals that set the initial radial compartment boundaries, but this has not been investigated. To measure gradient formation and morphogenetic precision in developing cochlea, we developed a quantitative image analysis procedure measuring SOX2 and pSMAD1/5/9 profiles in mouse embryos at embryonic day (E)12.5, E13.5, and E14.5. Intriguingly, we found that the pSMAD1/5/9 profile forms a linear gradient up to the medial ~ 75% of the PSD from the pSMAD1/5/9 peak in the lateral edge during E12.5 and E13.5. This is a surprising activity readout for a diffusive BMP4 ligand secreted from a tightly constrained lateral region since morphogens typically form exponential or power-law gradient shapes. This is meaningful for gradient interpretation because while linear profiles offer the theoretically highest information content and distributed precision for patterning, a linear morphogen gradient has not yet been observed. Furthermore, this is unique to the cochlear epithelium as the pSMAD1/5/9 gradient is exponential in the surrounding mesenchyme. In addition to the information-optimized linear profile, we found that while pSMAD1/5/9 is stable during this timeframe, an accompanying gradient of SOX2 shifts dynamically. Last, through joint decoding maps of pSMAD1/5/9 and SOX2, we see that there is a high-fidelity mapping between signaling activity and position in the regions that will become Kölliker’s organ and the organ of Corti. Mapping is ambiguous in the prosensory domain precursory to the outer sulcus. Altogether, this research provides new insights into the precision of early morphogenetic patterning cues in the radial cochlea prosensory domain.

Hedgehog morphogen gradient is robust towards variations in tissue morphology in Drosophila

During tissue development, gradients of secreted signaling molecules known as morphogens provide cells with positional information. The mechanisms underlying morphogen spreading have been widely studied, however, it remains largely unexplored whether the shape of morphogen gradients is influenced by tissue morphology. Here, we developed an analysis pipeline to quantify the distribution of proteins within a curved tissue. We applied it to the Hedgehog morphogen gradient in the Drosophila wing and eye-antennal imaginal discs, which are flat and curved tissues, respectively. Despite a different expression profile, the slope of the Hedgehog gradient was comparable between the two tissues. Moreover, inducing ectopic folds in wing imaginal discs did not affect the slope of the Hedgehog gradient. Suppressing curvature in the eye-antennal imaginal disc also did not alter the Hedgehog gradient slope but led to ectopic Hedgehog expression. In conclusion, through the development of an analysis pipeline that allows quantifying protein distribution in curved tissues, we show that the Hedgehog gradient is robust towards variations in tissue morphology.

Get me out of here: Sphingosine 1-phosphate signaling and T cell exit from tissues during an immune response.

During an immune response, the duration of T cell residence in lymphoid and non-lymphoid tissues likely affects T cell activation, differentiation, and memory development. The factors that govern T cell transit through inflamed tissues remain incompletely understood, but one important determinant of T cell exit from tissues is sphingosine 1-phosphate (S1P) signaling. In homeostasis, S1P levels are high in blood and lymph compared to lymphoid organs, and lymphocytes follow S1P gradients out of tissues into circulation using varying combinations of five G-protein coupled S1P receptors. During an immune response, both the shape of S1P gradients and the expression of S1P receptors are dynamically regulated. Here we review what is known, and key questions that remain unanswered, about how S1P signaling is regulated in inflammation and in turn how S1P shapes immune responses.

Optimal error estimates of the discrete shape gradients for shape optimizations governed by the Stokes-Brinkman equations

This work aims at investigating the convergence of shape gradients for shape optimizations governed by the Stokes-Brinkman equations. Two types of optimal control problems are considered, the shape inverse problem and the dissipated energy minimization problem, where the distinction between these two problems lies in the difference of the objective functionals. Lagrangian functional is introduced to obtain the adjoint equations and Eulerian derivatives at a fixed domain Ω in the direction V in volume and boundary integral forms are derived by the Piola material derivative approach and function space parametrization technique. Mixed finite element method is applied to discretize both the state and adjoint equations, as well as the corresponding Eulerian derivatives. The optimal error estimates for both forms of the shape derivatives are obtained. We infer that the volume-based shape derivative has a convergence rate of order h2, while the boundary-based one has a convergence rate of order h|log⁡h| under the MINI element. Numerical experiments are reported to demonstrate the theoretical analysis and indicate the better accuracy of volume-based expressions.

Phase-field study of polycrystalline growth and texture selection during melt pool solidification

Grain growth competition during solidification determines microstructural features, such as dendritic arm spacings, segregation pattern, and grain texture, which have a key impact on the final mechanical properties. During metal additive manufacturing (AM), these features are highly sensitive to manufacturing conditions, such as laser power and scanning speed. The melt pool (MP) geometry is also expected to have a strong influence on microstructure selection. Here, taking advantage of a computationally efficient multi-GPU implementation of a quantitative phase-field model, we use two-dimensional cross-section simulations of a shrinking MP during metal AM, at the scale of the full MP, in order to explore the resulting mechanisms of grain growth competition and texture selection. We explore MPs of different aspect ratios, different initial (substrate) grain densities, and repeat each simulation several times with different random grain distributions and orientations along the fusion line in order to obtain a statistically relevant picture of grain texture selection mechanisms. Our results show a transition from a weak to a strong ⟨10⟩ texture when the aspect ratio of the melt pool deviates from unity. This is attributed to the shape and directions of thermal gradients during solidification, and seems more pronounced in the case of wide melt pools than in the case of a deeper one. The texture transition was not found to notably depend upon the initial grain density along the fusion line from which the melt pool solidifies epitaxially.

Geometric Aspects of Shape Optimization

We present a review of known results in shape optimization from the point of view of Geometric Analysis. This paper is devoted to the mathematical aspects of the shape optimization theory. We focus on the theory of gradient flows of objective functions and their regularizations. Shape optimization is a part of calculus of variations which uses the geometry. Shape optimization is also related to the free boundary problems in the theory of Partial Differential Equations. We consider smooth perturbations of geometrical domains in order to develop the shape calculus for the analysis of shape optimization problems. There are many applications of such a framework, in solid and fluid mechanics as well as in the solution of inverse problems. For the sake of simplicity we consider model problems, in principle in two spatial dimensions. However, the methods presented are used as well in three spatial dimensions. We present a result on the convergence of the shape gradient method for a model problem. To our best knowledge it is the first result of convergence in shape optimization. The complete proofs of some results are presented in report (Plotnikov and Sokolowski, Gradient flow for Kohn–Vogelius functional).

Exploring the Bottom-Up Growth of Anisotropic Gold Nanoparticles from Substrate-Bound Seeds in Microfluidic Reactors.

We developed an unconventional seed-mediated in situ synthetic method, whereby gold nanostars are formed directly on the internal walls of microfluidic reactors. The dense plasmonic substrate coatings were grown in microfluidic channels with different geometries to elucidate the impacts of flow rate and profile on reagent consumption, product morphology, and density. Nanostar growth was found to occur in the flow-limited regime and our results highlight the possibility of creating shape gradients or incorporating multiple morphologies in the same microreactor, which is challenging to achieve with traditional self-assembly. The plasmonic-microfluidic platforms developed herein have implications for a broad range of applications, including cell culture/sorting, catalysis, sensing, and drug/gene delivery.

Study on heat transfer behavior and process optimization in differential phase electromagnetic DC casting of extra-large AZ31B alloy flat ingot: numerical simulation and experimental verification

Based on the heat transfer analysis in the primary cooling region of large-size magnesium alloy flat ingot Direct-chill (DC) casting, a novel crystallizer structure for flat ingot electromagnetic DC casting was put forward, in which the primary cooling can be controlled independently from the secondary cooling, and the circumferential control of the primary cooling can be realized synchronously. The design and optimization of the inner sleeve material and thickness and the distribution of the primary cooling were also carried out. For the AZ31B flat ingot with a cross-section size of 1200 × 400 mm, the copper inner sleeve material was selected, and it is beneficial to control temperature gradient and mushy zone shape in flat ingot DC casting. A uniform temperature distribution can be obtained when the cooling intensity ratio of the wide and narrow faces is 1:1/3. Furthermore, the influence of casting process parameters on the mushy zone characteristic of 1200 × 400 mm flat ingot was studied by orthogonal planning of process parameters and numerical simulation methods. Simultaneously, the casting process parameters of other flat ingot sizes were determined, and AZ31B alloy DC casting flat ingots with section sizes of 750 × 440 mm, 1200 × 400 mm, and 1600 × 400 mm under industrial conditions were carried out, and the crack-free preparation of the above billets was successfully achieved.

Shape sensitivity analysis of an elastic contact problem: Convergence of the Nitsche based finite element approximation

In a recent work, we introduced a finite element approximation for the shape optimization of an elastic structure in sliding contact with a rigid foundation where the contact condition (Signorini’s condition) is approximated by Nitsche’s method and the shape gradient is obtained via the adjoint state method. The motivation of this work is to propose an a priori convergence analysis of the numerical approximation of the variables of the shape gradient (displacement and adjoint state) and to show some numerical results in agreement with the theoretical ones. The main difficulty comes from the non-differentiability of the contact condition in the classical sense which requires the notion of conical differentiability.

Shape and topology optimization method for fiber placement design of CFRP plate and shell structures

In this study, we present a novel non-parametric shape-topology optimization method for fiber placement and orientation design, aiming at controlling the stiffness of CFRP (Carbon Fiber Reinforced Plastics) shell structures. The sum of squared error norm for achieving the target displacements on the arbitrarily specified position is minimized under the length constraints as a design problem. A simultaneous shape and topology optimization problem is formulated as a distributed-parameter optimization problem based on the variational method. The shape gradient function and the density gradient for this design optimization problem are theoretically derived with the Lagrange multiplier method, the material derivative method and the adjoint variable method. The solid isotropic material with penalization (SIMP) method is employed for topology optimization. The gradient functions derived are applied to the H1 gradient method for fibers to determine the optimal shape and topology. With the present method, the optimal fiber placement and orientation can be obtained without any parameterization. Numerical examples are solved to show the validity of the proposed method and the results are discussed.

Taxonomic Diversity and Selection of Functional Traits in Novel Ecosystems Developing on Coal-Mine Sedimentation Pools

Coal-mine sedimentation pools are extrazonal habitats in which the anthropogenic changes of all historic, abiotic, and biotic components, followed by conditions of extreme environmental stress, lead to the formation of novel ecosystems. Our study aims to (i) classify the vegetation on the basis of floristic and ecological criteria, (ii) detect the main environmental gradients responsible for the diversity of vegetation, and (iii) present the selection of species’ functional traits along environmental gradients. A cluster analysis of the floristic data revealed 14 distinct combinations of species. Short- and long-lived ruderals, meadow, xerothermic, and psammophilous species make up the floristic composition of vegetation. A canonical correspondence analysis on the floristic data and average Ellenberg’s indicator values confirmed moisture, soil reaction, and salinity as the main gradients, while fertility and insolation were secondary gradients shaping the diversity of vegetation. A RLQ with a subsequent cluster analysis revealed four groups of species traits selected along environmental gradients. These differed with reference to morphological (canopy height) and physiological traits (specific leaf area, or SLA), as well as persistence (life span), regeneration (reproduction by seeds or vegetative reproduction), and dispersal functional traits. This knowledge can be crucial when planning the restoration of these sites by using spontaneous succession and learning how the various environmental resources can be used to restore or provide new ecosystem services.

Ellenberg Indicator Values Disclose Complex Environmental Filtering Processes in Plant Communities along an Elevational Gradient

Ellenberg indicator values (EIVs) express plant preferences for temperature, light, continentality, soil moisture, pH, and soil nutrients, and have been largely used to deduce environmental characteristics from plant communities. However, EIVs might also be used to investigate the importance of filtering mechanisms in shaping plant communities according to species ecological preferences, a so far overlooked use of EIVs. In this paper, we investigated how community-weighted means (CWM), calculated with EIVs, varied along an elevational gradient in a small mountain in Central Italy. We also tested if species abundances varied according to their ecological preferences. We found that the prevalence of thermophilous species declines with elevation, being progressively replaced by cold-adapted species. Heliophilous species prevail at low and high elevations (characterized by the presence of open habitats), whereas in the middle of the gradient (occupied by the beech forest), sciophilous species predominate. Variations for moisture and soil nutrient preferences followed a similar pattern, probably because of the high moisture and nutrient levels of forest soils with a lot of humus. No distinct pattern was detected for EIVs for pH and continentality since these factors are subject to more local variations. These results highlight the possible role of EIVs to investigate how environmental gradients shape plant communities.

The Gaia-ESO survey: Mapping the shape and evolution of the radial abundance gradients with open clusters

Context. The spatial distribution of elemental abundances and their time evolution are among the major constraints to disentangling the scenarios of formation and evolution of the Galaxy. Aims. In this paper we used the sample of open clusters available in the final release of the Gaia-ESO survey to trace the Galactic radial abundance and abundance-to-iron ratio gradients, and their time evolution. Methods. We selected member stars in 62 open clusters, with ages from 0.1 to about 7 Gyr, located in the Galactic thin disc at galactocentric radii (RGC) from about 6 to 21 kpc. We analysed the shape of the resulting [Fe/H] gradient, the average gradients [El/H] and [El/Fe] combining elements belonging to four different nucleosynthesis channels, and their individual abundance and abundance ratio gradients. We also investigated the time evolution of the gradients dividing open clusters in three age bins. Results. The [Fe/H] gradient has a slope of −0.054 dex kpc−1. It can be better approximated with a two-slope shape, steeper for RGC ≤ 11.2 kpc and flatter in the outer regions. We saw different behaviours for elements belonging to different channels. For the time evolution of the gradient, we found that the youngest clusters (age < 1 Gyr) in the inner disc have lower metallicity than their older counterparts and that they outline a flatter gradient. We considered some possible explanations, including the effects of gas inflow and migration. We suggest that the most likely one may be related to a bias introduced by the standard spectroscopic analysis producing lower metallicities in the analysis of low-gravity stars. Conclusions. To delineate the shape of the ‘true’ gradient, we should most likely limit our analysis to stars with low surface gravity log g > 2.5 and microturbulent parameter ξ < 1.8 km s−1. Based on this reduced sample, we can conclude that the gradient has minimally evolved over the time-frame outlined by the open clusters, indicating a slow and stationary formation of the thin disc over the last 3 Gyr. We found a secondary role of cluster migration in shaping the gradient, with a more prominent role of migration for the oldest clusters.

Modelling, analysis, and numerical methods for a geometric inverse source problem in variable-order time-fractional subdiffusion

There exist research works on studying time-dependent integer-order and time-fractional constant-order geometric inverse source problems in the literature. The time-fractional variable-order geometric inverse source problems although also have important physical applications have not been studied mathematically and numerically in literature. The aim of this work is to study an inverse source problem associated with a variable-order time-fractional subdiffusion equation. We first build a mathematical model and show existence of the optimal shape for shape reconstruction of the source support. Then, shape sensitivity analysis is performed to propose a shape gradient optimization algorithm allowing deformations for numerically solving the model problem. In order to reconstruct the source support with topology unknown a priori, moreover, we build a phase-field model and propose a gradient algorithm allowing both shape and topological changes by a phase-field method. A variety of numerical examples are presented to demonstrate effectiveness of the two algorithms.

On the new coupled complex boundary method in shape optimization framework for solving stationary free boundary problems

<p style='text-indent:20px;'>We expose here a novel application of the so-called coupled complex boundary method – first put forward by Cheng et al. (2014) to deal with inverse source problems – in the framework of shape optimization for solving the exterior Bernoulli problem, a prototypical model of stationary free boundary problems. The idea of the method is to transform the overdetermined problem to a complex boundary value problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary conditions on the free boundary. Then, we optimize the cost function constructed by the imaginary part of the solution in the whole domain in order to identify the free boundary. We also prove the existence of the shape derivative of the complex state with respect to the domain. Afterwards, we compute the shape gradient of the cost functional, and characterize its shape Hessian at the optimal domain under a strong, and then a mild regularity assumption on the domain. We then prove the ill-posedness of the proposed shape problem by showing that the latter expression is compact. Also, we devise an iterative algorithm based on a Sobolev gradient scheme via finite element method to solve the minimization problem. Finally, we illustrate the applicability of the method through several numerical examples, both in two and three spatial dimensions.</p>

On Finite Element Approximations to a Shape Gradient Flow in Shape Optimization of Elliptic Problems

On Finite Element Approximations to a Shape Gradient Flow in Shape Optimization of Elliptic Problems

Shape optimization method for strength design problem of microstructures in a multiscale structure

AbstractIn this paper, we propose a solution to a shape optimization problem for the strength design of periodic microstructures in multiscale structures. Two maximum stress minimization problems are addressed: minimization of maximum microstructural stress and minimization of maximum macrostructural stress. The homogenization method is used to bridge the macrostructure and the microstructure and to calculate local microstructural stress. By replacing the maximum stress value with a Kreisselmeier‐Steinhauser function, the difficulty of nondifferentiability of maximum stress is avoided. Each strength design problem is formulated as a distributed parameter optimization problem subject to an area constraint including the whole microstructure. The shape gradient functions for both problems are derived using Lagrange's undetermined multiplier method, the material derivative method, and the adjoint variable method. The H1 gradient method is used to determine the unit cell shapes of the microstructure, while reducing the objective function and maintaining smooth design boundaries. In the numerical examples, the optimal shapes obtained for minimization of the maximum local stress of the microstructure and the macrostructure are compared and discussed. The results confirm the effectiveness of the microstructure shape optimization method for the two strength design problems of multiscale structures.