scholarly journals Fireshape: a shape optimization toolbox for Firedrake

2021 ◽  
Vol 63 (5) ◽  
pp. 2553-2569
Author(s):  
Alberto Paganini ◽  
Florian Wechsung
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Shape Optimization ◽  
Open Source ◽  
Optimization Problems ◽  
Moving Mesh ◽  
Moving Mesh Method ◽  
Finite Element Software ◽  
Mesh Method

AbstractWe introduce Fireshape, an open-source and automated shape optimization toolbox for the finite element software Firedrake. Fireshape is based on the moving mesh method and allows users with minimal shape optimization knowledge to tackle with ease challenging shape optimization problems constrained to partial differential equations (PDEs).

scholarly journals Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

2011 ◽  
Vol 10 (3) ◽  
pp. 509-576 ◽  
Author(s):  
M. J. Baines ◽  
M. E. Hubbard ◽  
P. K. Jimack
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Historical Review ◽  
Porous Medium Equation ◽  
Moving Mesh ◽  
Test Problems ◽  
Partial Differential ◽  
Two Phase

AbstractThis article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

Swelling Characteristics of 3D-Arbitrary-Geometry of the pH-Sensitive Hydrogels

2013 ◽  
Author(s):  
Kamlesh J. Suthar ◽  
Derrick C. Mancini ◽  
Muralidhar K. Ghantasala
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Material Properties ◽  
Ph Sensitive ◽  
Moving Mesh Method ◽  
Partial Differential ◽  
Highly Nonlinear ◽  
Simulation Results ◽  
Ph Sensitive Hydrogel

We present our simulation results of swelling responses of the pH-sensitive, 3D-arbitarary-geometry hydrogel in steady state conditions. The swelling responses of the hydrogels to the changes in environmental stimuli such as solution pH are discussed. The finite element simulation uses three nonlinear partial-differential equations for responsible physical phenomena namely- chemical for ionic transport across the hydrogel, electrical for local electric charge balance within hydrogel, and mechanical for expansion of the hydrogel by the Nernst-Planck, the Poisson’s, and the mechanical field equations respectively. In the case of pH-sensitive hydrogel, material properties such as modulus of elasticity and Poisson’s ratio changes with a change in surrounding environments. Finite element analysis used for present study was carried out by full coupling of above three partial-differential equations with variable material properties. Employing a moving mesh method for 3D geometry, the FEM simulation was performed to account for large-swelling of the pH-sensitive hydrogel. This highly nonlinear and computationally intensive simulation was performed using multicore parallel-processing computer. The simulation results using above mentioned strategy has been validated for 2D geometry and results are in agreement with other published experimental results.

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