On the shape gradient and shape Hessian of a shape functional subject to Dirichlet and Robin conditions

2014 ◽  
Vol 8 ◽  
pp. 5387-5397
Author(s):  
Jerico B. Bacani
Keyword(s):  
Shape Gradient ◽  
Shape Hessian

scholarly journals Computing the shape gradient of stellarator coil complexity with respect to the plasma boundary

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
Arthur Carlton-Jones ◽  
Elizabeth J. Paul ◽  
William Dorland
Keyword(s):  
Null Space ◽  
Radial Distance ◽  
Plasma Boundary ◽  
Optimization Approach ◽  
Surface Parameters ◽  
Fixed Boundary ◽  
Plasma Surface ◽  
Shape Gradient ◽  
Derivative Free ◽  
Boundary Optimization

Coil complexity is a critical consideration in stellarator design. The traditional two-step optimization approach, in which the plasma boundary is optimized for physics properties and the coils are subsequently optimized to be consistent with this boundary, can result in plasma shapes which cannot be produced with sufficiently simple coils. To address this challenge, we propose a method to incorporate considerations of coil complexity in the optimization of the plasma boundary. Coil complexity metrics are computed from the current potential solution obtained with the REGCOIL code (Landreman, Nucl. Fusion, vol. 57, 2017, 046003). While such metrics have previously been included in derivative-free fixed-boundary optimization (Drevlak et al., Nucl. Fusion, vol. 59, 2018, 016010), we compute the local sensitivity of these metrics with respect to perturbations of the plasma boundary using the shape gradient (Landreman & Paul, Nucl. Fusion, vol. 58, 2018, 076023). We extend REGCOIL to compute derivatives of these metrics with respect to parameters describing the plasma boundary. In keeping with previous research on winding surface optimization (Paul et al., Nucl. Fusion, vol. 58, 2018, 076015), the shape derivatives are computed with a discrete adjoint method. In contrast with the previous work, derivatives are computed with respect to the plasma surface parameters rather than the winding surface parameters. To further reduce the resolution required to compute the shape gradient, we present a more efficient representation of the plasma surface which uses a single Fourier series to describe the radial distance from a coordinate axis and a spectrally condensed poloidal angle. This representation is advantageous over the standard cylindrical representation used in the VMEC code (Hirshman & Whitson, Phys. Fluids, vol. 26, 1983, pp. 3553–3568), as it provides a uniquely defined poloidal angle, eliminating a null space in the optimization of the plasma surface. In comparison with previous spectral condensation methods (Hirshman & Breslau, Phys. Plasmas, vol. 5, 1998, p. 2664), the modified poloidal angle is obtained algebraically rather than through the solution of a nonlinear optimization problem. The resulting shape gradient highlights features of the plasma boundary that are consistent with simple coils and can be used to couple coil and fixed-boundary optimization.

scholarly journals Adjoint approach to calculating shape gradients for three-dimensional magnetic confinement equilibria

2019 ◽  
Vol 85 (2) ◽  
Author(s):  
Thomas Antonsen ◽  
Elizabeth J. Paul ◽  
Matt Landreman
Keyword(s):  
Figure Of Merit ◽  
Transport Coefficients ◽  
Three Dimensional ◽  
Direct Calculation ◽  
Flux Surface ◽  
Shape Gradient ◽  
Onsager Symmetry ◽  
Mhd Equilibria ◽  
Gradient Based ◽  
Adjoint Approach

The shape gradient quantifies the change in some figure of merit resulting from differential perturbations to a shape. Shape gradients can be applied to gradient-based optimization, sensitivity analysis and tolerance calculation. An efficient method for computing the shape gradient for toroidal three-dimensional magnetohydrodynamic (MHD) equilibria is presented. The method is based on the self-adjoint property of the equations for driven perturbations of MHD equilibria and is similar to the Onsager symmetry of transport coefficients. Two versions of the shape gradient are considered. One describes the change in a figure of merit due to an arbitrary displacement of the outer flux surface; the other describes the change in the figure of merit due to the displacement of a coil. The method is implemented for several example figures of merit and compared with direct calculation of the shape gradient. In these examples the adjoint method reduces the number of equilibrium computations by factors of$O(N)$, where$N$is the number of parameters used to describe the outer flux surface or coil shapes.

Dynamical shape gradient for the Navier–Stokes system

2004 ◽  
Vol 338 (2) ◽  
pp. 183-186 ◽  
Author(s):  
Raja Dziri ◽  
Marwan Moubachir ◽  
Jean-Paul Zolésio
Keyword(s):  
Stokes System ◽  
Navier Stokes ◽  
Shape Gradient

Shape-gradient optimization applied to the reconstruction of 2-D and 3-D metallic objects

2009 ◽  
Author(s):  
Ralph Ferraye ◽  
Pierre Dubois ◽  
Ioannis Aliferis ◽  
Jean-Yves Dauvignac ◽  
Christian Pichot ◽  
...  
Keyword(s):  
Shape Gradient ◽  
Gradient Optimization

scholarly journals Shape Gradient for Isogeometric Structural Design

2013 ◽  
Vol 161 (2) ◽  
pp. 361-367 ◽  
Author(s):  
Louis Blanchard ◽  
Régis Duvigneau ◽  
Anh-Vu Vuong ◽  
Bernd Simeon
Keyword(s):  
Structural Design ◽  
Shape Gradient

Extension of covariant derivative (III): From classical gradient to shape gradient

2015 ◽  
Vol 31 (1) ◽  
pp. 96-103 ◽  
Author(s):  
Ya-Jun Yin
Keyword(s):  
Covariant Derivative ◽  
Shape Gradient

scholarly journals Free-Form Optimization of Thin-Walled Structure for Frequency Response Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Masatoshi Shimoda ◽  
Yang Liu
Keyword(s):  
Frequency Response ◽  
Optimization Method ◽  
Free Form ◽  
Error Norm ◽  
Material Derivative ◽  
Shape Gradient ◽  
Thin Walled Structures ◽  
Target Values ◽  
Gradient Function ◽  
Thin Walled

We present a node-based free-form optimization method for designing forms of thin-walled structures in order to control vibration displacements or mode at a prescribed frequency. A squared displacement error norm is introduced at the prescribed surface as the objective functional to control the vibration displacements to target values in a frequency response problem. It is assumed that the thin-walled structure is varied in the normal direction to the surface and the thickness is constant. A nonparametric shape optimization problem is formulated, and the shape gradient function is theoretically derived using the material derivative method and the adjoint variable method. The shape gradient function obtained is applied to the surface of the thin-walled structure as a fictitious traction force to vary the form. With this free-form optimization method, an optimum thin-walled structure with a smooth free-form surface can be obtained without any shape parameterization. The calculated results show the effectiveness of the proposed method for the optimal free-form design of thin-walled structures with vibration mode control.

Shape Identification for Controlling the Static Deformation of Frame Structures

2014 ◽  
Author(s):  
Koki Kameyama ◽  
Masatoshi Shimoda ◽  
Takashi Morimoto
Keyword(s):  
Optimization Method ◽  
Frame Structure ◽  
Static Deformation ◽  
Free Form ◽  
Frame Structures ◽  
Shape Identification ◽  
Shape Gradient ◽  
Deformation Control ◽  
Out Of Plane ◽  
Gradient Function

The deformation control is an important design problem in the stiffness design of structures and it also enables to give a function to the structures. This paper proposes a non-parametric, or a node-based shape optimization method based on the variational method for controlling the static deformation of spatial frame structures. As the objective functional, we introduce the sum of squared error norms to the desired displacements on specified members. Under the assumption that each member varies in the out-of-plane direction to the centroidal axis, the shape gradient function and the optimality conditions are theoretically derived. The shape gradient function is applied to a gradient method in a function space with a Laplacian smoother. With this method, an optimal free-form frame structure with smoothness can be identified for a desired static deformation. The validity and effectiveness were verified through design examples.

An ensemble shape gradient features descriptor based nodule detection paradigm: a novel model to augment complex diagnostic decisions assistance

2018 ◽  
Vol 79 (13-14) ◽  
pp. 8649-8675 ◽  
Author(s):  
M. Arfan Jaffar ◽  
M. Sultan Zia ◽  
Majid Hussain ◽  
Abdul Basit Siddiqui ◽  
Sheeraz Akram ◽  
...  
Keyword(s):  
Shape Gradient ◽  
Nodule Detection ◽  
Novel Model
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