Shape Sensitivity Research Articles

Shape Differentiability of Lagrangians and Application to Stokes Problem

A class of convex constrained minimization problems over polyhedral cones for geometry-dependent quadratic objective functions is considered in a functional analysis framework. Shape differentiability of the primal minimization problem needs a bijective property for mapping of the primal cone. This restrictive assumption is relaxed to bijection of the dual cone within the Lagrangian formulation as a primal-dual minimax problem. In this paper, we give results on primal-dual shape sensitivity analysis that extends the class of shape-differentiable problems supported by explicit formula of the shape derivative. We apply the results to the Stokes problem under mixed Dirichlet-Neumann boundary conditions subject to the divergence-free constraint.

Shape sensitivity for a two-phase heat conduction problem considering nanoscale effects

Shape sensitivity for a two-phase heat conduction problem considering nanoscale effects

A development of isogeometric boundary element method for 3D acoustics and its application to shape sensitivity analysis

A development of isogeometric boundary element method for 3D acoustics and its application to shape sensitivity analysis

Shape optimization using structural adjoint and RBF mesh morphing

Adjoint solvers are acquiring nowadays a growing importance in shape optimization especially when dealing with fluid dynamic applications; their use for structural optimization is however still limited. In this work an optimization workflow based on the synergic use of a structural continuum-discrete adjoint variable solver and the commercial morpher RBF Morph™ is presented. Shape sensitivity information with respect to the objective function is exported as deformation maps on the interested geometry and transferred to the morpher that, after a proper filtering and setup, allows to update automatically the numerical grid. By employing a gradient based logic it is finally possible to achieve an evolutionary optimization. The proposed method effectiveness is shown with two examples: a cantilever beam and a structural bracket.

Structural shape optimization using Cartesian grids and automatic h-adaptive mesh projection

We present a novel approach to 3D structural shape optimization that leans on an Immersed Boundary Method. A boundary tracking strategy based on evaluating the intersections between a fixed Cartesian grid and the evolving geometry sorts elements as internal, external and intersected. The integration procedure used by the NURBS-Enhanced Finite Element Method accurately accounts for the nonconformity between the fixed embedding discretization and the evolving structural shape, avoiding the creation of a boundary-fitted mesh for each design iteration, yielding in very efficient mesh generation process. A Cartesian hierarchical data structure improves the efficiency of the analyzes, allowing for trivial data sharing between similar entities or for an optimal reordering of the matrices for the solution of the system of equations, among other benefits. Shape optimization requires the sufficiently accurate structural analysis of a large number of different designs, presenting the computational cost for each design as a critical issue. The information required to create 3D Cartesian h-adapted mesh for new geometries is projected from previously analyzed geometries using shape sensitivity results. Then, the refinement criterion permits one to directly build h-adapted mesh on the new designs with a specified and controlled error level. Several examples are presented to show how the techniques here proposed considerably improve the computational efficiency of the optimization process.

An extension of shape sensitivity analysis to an immersed boundary method based on Cartesian grids

Gradient-based shape optimization processes of mechanical components require the gradients (sensitivity) of the magnitudes of interest to be calculated with sufficient accuracy. The aim of this study was to develop algorithms for the calculation of shape sensitivities considering geometric representation by parametric surfaces (i.e. NURBS or T-splines) using 3D Cartesian h-adapted meshes independent of geometry. A formulation of shape sensitivities was developed for an environment based on Cartesian meshes independent of geometry, which implies, for instance, the need to take into account the special treatment of boundary conditions imposed in non body-fitted meshes. The immersed boundary framework required to implement new methods of velocity field generation, which have a primary role in the integration of both the theoretical concepts and the discretization tools in shape design optimization. Examples of elastic problems with three-dimensional components are given to demonstrate the efficiency of the algorithms.

Modeling and optimal design of fiber-optic probe for medium detection based on computer simulation

This paper conducted a deep study on the technique of fiber-optic probe (FOP). Firstly, the feasibility of FOP medium detection was analyzed by establishing a mathematic model. Secondly, in order to improve the sensitivity of FOP, the material, shape and angle of conic sensitive tip of FOP were optimally designed by analyzing simulation experiments. Besides for improving light transmitting efficiency of FOP, based on simulation experiments, the optimum structure was concluded by evaluating the performances of eight structures of the bundle coupling light path. Lastly, a sapphire-FOP was developed on the foundation of above optimizations, the sapphire-FOP has high sensitivity and compact size, meanwhile, the output response experiment of the sapphire-FOP in different media, the response characteristics experiments of the sapphire-FOP piercing a gas bubble, and the response characteristics experiments of detection multiple gas bubbles in water medium were carried out to identify the performance of the sapphire-FOP in gas resolution, anti-contamination and response sensitivity.

A level set method for shape optimization in semilinear elliptic problems

We develop a finite-element based level set method for numerically solve shape optimization problems constrained by semilinear elliptic problems. By combining the shape sensitivity analysis and level set method, a gradient descent algorithm is proposed to solve the model problem. Different from solving the nonlinear Hamilton–Jacobi equations with finite differences in traditional level set methods, we solve the linear convection equation and reinitialization equation using the characteristic Galerkin finite element method. The methodology can handle topology as well as shape changes in both regular and irregular design regions. Numerical results are presented to demonstrate the effectiveness of our algorithm as well as to verify symmetry preserving and breaking properties of optimal subdomains.

Shape optimization of sound barrier using an isogeometric fast multipole boundary element method in two dimensions

This study presents an isogeometric fast multipole boundary element method (IGA FMBEM) in two-dimensional (2D) acoustics and a related sensitivity-based shape optimization algorithm for sound barriers. In the isogeometric analysis, Non-Uniform Rational B-Splines (NURBS) are used to accurately represent structural geometry. The control points are set as design variables in the shape optimization procedure given that their variations can flexibly result in shape changes. Acoustic shape sensitivities with respect to control points are calculated by the sensitivity boundary integral equation (BIE) based on the direct differentiation method. The singular integrals in the sensitivity BIEs are formulated explicitly under the isogeometric discretization. The minimization of sound pressure on the reference surface is selected as design objective. The gradient-based optimization solver is finally introduced for optimization iteration after the acoustic state and sensitivity information are obtained. The fast multipole method (FMM) is applied to improve overall computational efficiency. The Burton–Miller method is adopted to conquer the fictitious eigenfrequency problem in solving exterior acoustic problems. The correctness and validity of the proposed methods are demonstrated through a number of numerical simulations, while the performance of the sensitivity-based optimization algorithm is observed in the shape optimization of a 2D Γ-shaped sound barrier.

Inclusion boundary reconstruction and sensitivity analysis in electrical impedance tomography

Reconstruction of conductive inclusions in a homogeneous background medium is commonly seen in electrical impedance tomography (EIT). One of the methods to deal with the inclusion reconstruction problems is the shape-based method. With prior knowledge of conductivity of target inclusions, the boundary of inclusions is parameterized by several shape coefficients and recovered from EIT measurements. This paper presents a shape-based inclusion reconstruction method and its numerical implementation with boundary element method (BEM). A shape perturbation method (SPM) is proposed to calculate the shape sensitivity in EIT. To evaluate the accuracy of the presented method, a series of numerical tests are conducted. The characteristics of EIT shape sensitivity are analysed. Some factors influencing the reconstruction performance are discussed.

The large-scale organization of shape processing in the ventral and dorsal pathways.

Although shape perception is considered a function of the ventral visual pathway, evidence suggests that the dorsal pathway also derives shape-based representations. In two psychophysics and neuroimaging experiments, we characterized the response properties, topographical organization and perceptual relevance of these representations. In both pathways, shape sensitivity increased from early visual cortex to extrastriate cortex but then decreased in anterior regions. Moreover, the lateral aspect of the ventral pathway and posterior regions of the dorsal pathway were sensitive to the availability of fundamental shape properties, even for unrecognizable images. This apparent representational similarity between the posterior-dorsal and lateral-ventral regions was corroborated by a multivariate analysis. Finally, as with ventral pathway, the activation profile of posterior dorsal regions was correlated with recognition performance, suggesting a possible contribution to perception. These findings challenge a strict functional dichotomy between the pathways and suggest a more distributed model of shape processing.

Stress-based shape and topology optimization with the level set method

This paper proposes a level set method to solve minimum stress and stress-constrained shape and topology optimization problems. The method solves a sub-optimization problem every iteration to obtain optimal boundary velocities. A p-norm stress functional is used to aggregate stresses in a single constraint. The shape sensitivity function is derived and a computational procedure based on a least squares interpolation approach is devised in order to compute sensitivities at the boundaries. Adaptive constraint scaling is used to enforce exact control of stress limits. Numerical results show that the method is able to solve the problem efficiently for single and multiple load cases obtaining solutions with smooth boundaries.

Sensitivity analysis of a shape control problem for the Navier--Stokes equations

We deal with a sensitivity analysis of an optimal shape control problem for the stationary Navier-Stokes system. A two-dimensional channel flow of an incompressible, viscous fluid is examined to determine the shape of a bump on a part of the boundary that minimizes the viscous drag. By using the material derivative method and adjoint variables for a shape sensitivity analysis, we derive the shape gradient of the design functional for the model problem.

Topology optimization of hyperelastic structures using a level set method

Soft rubberlike materials, due to their inherent compliance, are finding widespread implementation in a variety of applications ranging from assistive wearable technologies to soft material robots. Structural design of such soft and rubbery materials necessitates the consideration of large nonlinear deformations and hyperelastic material models to accurately predict their mechanical behaviour. In this paper, we present an effective level set-based topology optimization method for the design of hyperelastic structures that undergo large deformations. The method incorporates both geometric and material nonlinearities where the strain and stress measures are defined within the total Lagrange framework and the hyperelasticity is characterized by the widely-adopted Mooney–Rivlin material model. A shape sensitivity analysis is carried out, in the strict sense of the material derivative, where the high-order terms involving the displacement gradient are retained to ensure the descent direction. As the design velocity enters into the shape derivative in terms of its gradient and divergence terms, we develop a discrete velocity selection strategy. The whole optimization implementation undergoes a two-step process, where the linear optimization is first performed and its optimized solution serves as the initial design for the subsequent nonlinear optimization. It turns out that this operation could efficiently alleviate the numerical instability and facilitate the optimization process. To demonstrate the validity and effectiveness of the proposed method, three compliance minimization problems are studied and their optimized solutions present significant mechanical benefits of incorporating the nonlinearities, in terms of remarkable enhancement in not only the structural stiffness but also the critical buckling load.

Residual stress sensitivity analysis using a complex variable finite element method

The application of the complex variable finite element method, ZFEM, to structures containing residual stresses is detailed. A thick-walled sphere model is subjected to an autofrettage process to induce the residual stress field and solved using ZFEM. The stress results from ZFEM are compared to an analytical solution and to standard finite element results. Sensitivities are calculated with respect to material parameters and to the applied load. The ZFEM partial derivative results are shown to provide excellent accuracy when compared with numerical derivatives of the analytical solution and finite differencing of standard finite element solutions. Hence, ZFEM provides an effective step-size independent method to compute shape, material and loading sensitivities for structures subjected to residual stresses.

Drag reduction of slender blunt-based bodies using optimized rear cavities

We have investigated the use of the adjoint sensitivity formulation to design efficient passive control strategies aiming at reducing the drag coefficient of a slender blunt-based body with a straight rear cavity. In particular, two control techniques consisting in wake modifications generated by placing small control cylinders in the near wake and by geometry variations of the cavity respectively, have been evaluated numerically. Thus, we have computed the turbulent flow sensitivity of the drag coefficient to localized forcing for a two-dimensional body with a straight cavity at Re=ρU∞H∕μ=2000, where U∞ is the free-stream velocity, ρ and μ the fluid density and viscosity respectively and H the body height, showing that the highest values of sensitivity are obtained near the cavity edges. The effect of placing a pair of control cylinders around the most sensitive locations has been studied, obtaining a largest drag reduction of only 0.6%. Alternatively, a most efficient control strategy based on shape optimization has been thoroughly investigated. The drag shape sensitivity on the body surface (Othmer, 2014), computed using the former linear adjoint formulation, has been used in combination with a free-form deformation algorithm (Han et al., 2011), to guide the local structure deformations of the cavity, providing progressive drag reductions until the optimal, curved, shape is achieved. To deeply analyze the physical mechanisms behind the drag reduction provided by the optimal cavity, we have also performed more realistic three-dimensional numerical simulations using an IDDES model at two different Reynolds numbers, Re=2000 and 20000. The results corroborate the sensitivity analysis, obtaining a total drag reduction of 25.6% at Re=2000 and 43.9% at Re=20000, with respect to the original body without cavity, and 21.7% at Re=2000 and 29.6% at Re=20000 with respect to the body with a straight cavity. These reductions are mainly achieved by the inwards deflection of the flow upon detachment and a flow deceleration at the trailing edge due to an adverse pressure gradient introduced by the curved shape of the optimal cavity walls. Both combined effects reduce the size of the recirculation bubble formed behind the body, increasing the base pressure, and consequently, decreasing the drag. Furthermore, the addition of the optimized base cavity reduces the amplitude of the velocity fluctuations behind the body and stabilizes the wake, which becomes less chaotic and more two-dimensional.

Dot Sensitivity Analysis for Topology Optimization of Dielectric Material in Electrostatic System

In this paper, a dot sensitivity analysis is proposed for the topology optimization of dielectric materials in an electrostatic system. Dot sensitivity formula is analytically derived in closed form using the continuum shape sensitivity formula. The level set method is used for evolving the geometry in the topology optimization. As the dot sensitivity analysis can generate new material in a vacant region, it does not require an initial design for the dielectric material and widens the design space. Two numerical examples are tested to show the usefulness of dot sensitivity analysis.

Dosimetric impact assessment using a general algorithm in geant4 simulations for a complex-shaped multileaf collimator

PurposeWe have developed an inhouse algorithm for the multileaf collimator (MLC) geometry model construction with an appropriate accuracy for dosimetric tests. Our purpose is to build a complex type of MLC and analyze the influence of the modeling parameters on the dose calculation. MethodsUsing radiochromic films as detector the following tests were done: (I) Density test field: to compare measured and calculated dose distributions in order to determine the tungsten alloy physical density value. (II) Leaf ends test field: to verify the penumbra shape sensitivity against the discretization level set to simulate the curved leaf ends. (III) MLC-closed field: to obtain the value of the air gap between opposite leaves for a closed configuration which completes the modeling of the MLC leakage radiation. (IV) Picket-fence field: to fit the leaf tilt angle with respect of the divergent ray emerging from the source. ResultsFor a 18.5g/cm3 density value we have obtained a maximum, minimum and mean leakage values of 0.43%, 0.36% and 0.38%, similar to the experimental ones. The best discretization level in the leaf ends field shows a 5.51mm FWHM, very close to the measured value (5.49mm). An air gap of 370μm has been used in the simulation for the separation between opposite leaves. Using a 0.44° tilt angle, we found the same pattern as the experimental values. ConclusionsOur code can reproduce complex MLC designs with a submilimetric dosimetric accuracy which implies the necessary background for dose calculation of high clinical interest small fields.

SAMPA Chip: the New 32 Channels ASIC for the ALICE TPC and MCH Upgrades

This paper presents the test results of the second prototype of SAMPA, the ASIC designed for the upgrade of read-out front end electronics of the ALICE Time Projection Chamber (TPC) and Muon Chamber (MCH). SAMPA is made in a 130 nm CMOS technology with 1.25 V nominal voltage supply and provides 32 channels, with selectable input polarity, and three possible combinations of shaping time and sensitivity. Each channel consists of a Charge Sensitive Amplifier, a semi-Gaussian shaper and a 10-bit ADC; a Digital Signal Processor provides digital filtering and compression capability. In the second prototype run both full chip and single test blocks were fabricated, allowing block characterization and full system behaviour studies. Experimental results are here presented showing agreement with requirements for both the blocks and the full chip.

Own attractiveness and perceived relationship quality shape sensitivity in women’s memory for other men on the attractiveness dimension

Although recent work suggests that opposite-sex facial attractiveness is less salient in memory when individuals are in a committed romantic relationship, romantic relationship quality can vary over time. In light of this, we tested whether activating concerns about romantic relationship quality strengthens memory for attractive faces. Partnered women were exposed briefly to faces manipulated in shape cues to attractiveness before either being asked to think about a moment of emotional closeness or distance in their current relationship. We measured sensitivity in memory for faces as the extent to which they recognized correct versions of studied faces over versions of the same person altered to look either more or less-attractive than their original (i.e., studied) version. Contrary to predictions, high relationship quality strengthened hit rate for faces regardless of the sex or attractiveness of the face. In general, women’s memories were more sensitive to attractiveness in women, but were biased toward attractiveness in male faces, both when responding to unfamiliar faces and versions of familiar faces that were more attractive than the original male identity from the learning phase. However, findings varied according to self-rated attractiveness and a psychometric measure of the quality of their current relationship. Attractive women were more sensitive to attractiveness in men, while their less-attractive peers had a stronger bias to remember women as more-attractive and men as less-attractive than their original image respectively. Women in better-quality romantic relationships had stronger positive biases toward, and false memories for, attractive men. Our findings suggest a sophisticated pattern of sensitivity and bias in women’s memory for facial cues to quality that varies systematically according to factors that may alter the costs of female mating competition (‘market demand’) and relationship maintenance.