scholarly journals A novel [[EQUATION]] approach to shape optimisation with Lipschitz domains

2021 ◽  
Author(s):  
Klaus Deckelnick ◽  
Philip Herbert ◽  
Michael Hinze
Keyword(s):  
Optimal Transport ◽  
Existence Of Solutions ◽  
Specific Form ◽  
Steepest Descent ◽  
Lipschitz Domains ◽  
Shape Optimisation ◽  
Shape Derivative ◽  
Optimal Shapes ◽  
Novel Method ◽  
Descent Directions

This article introduces a novel method for the implementation of shape optimisation with Lipschitz domains. We propose to use the shape derivative to determine deformation fields which represent steepest descent directions of the shape functional in the $W^{1,\infty}-$ topology. The idea of our approach is demonstrated for shape optimisation of $n$-dimensional star-shaped domains, which we represent as functions defined on the unit $(n-1)$-sphere. In this setting we provide the specific form of the shape derivative and prove the existence of solutions to the underlying shape optimisation problem. Moreover, we show the existence of a direction of steepest descent in the $W^{1,\infty}-$ topology. We also note that shape optimisation in this context is closely related to the $\infty-$Laplacian, and to optimal transport, where we highlight the latter in the numerics section. We present several numerical experiments illustrating that our approach seems to be superior over existing Hilbert space methods, in particular in developing optimal shapes with corners.

scholarly journals A novel p-harmonic descent approach applied to fluid dynamic shape optimization

2021 ◽  
Author(s):  
Peter Marvin Müller ◽  
Niklas Kühl ◽  
Martin Siebenborn ◽  
Klaus Deckelnick ◽  
Michael Hinze ◽  
...  
Keyword(s):  
Shape Optimization ◽  
Large Deformations ◽  
Fluid Dynamic ◽  
Shape Optimisation ◽  
Shape Features ◽  
Floating Bodies ◽  
Novel Method ◽  
Sharp Corners ◽  
Dynamic Shape

AbstractWe introduce a novel method for the implementation of shape optimization for non-parameterized shapes in fluid dynamics applications, where we propose to use the shape derivative to determine deformation fields with the help of the $$p-$$ p - Laplacian for $$p > 2$$ p > 2 . This approach is closely related to the computation of steepest descent directions of the shape functional in the $$W^{1,\infty }-$$ W 1 , ∞ - topology and refers to the recent publication Deckelnick et al. (A novel $$W^{1,\infty}$$ W 1 , ∞ approach to shape optimisation with Lipschitz domains, 2021), where this idea is proposed. Our approach is demonstrated for shape optimization related to drag-minimal free floating bodies. The method is validated against existing approaches with respect to convergence of the optimization algorithm, the obtained shape, and regarding the quality of the computational grid after large deformations. Our numerical results strongly indicate that shape optimization related to the $$W^{1,\infty }$$ W 1 , ∞ -topology—though numerically more demanding—seems to be superior over the classical approaches invoking Hilbert space methods, concerning the convergence, the obtained shapes and the mesh quality after large deformations, in particular when the optimal shape features sharp corners.

scholarly journals Solutions of a Class of Degenerate Kinetic Equations Using Steepest Descent in Wasserstein Space

2020 ◽  
Vol 2020 ◽  
pp. 1-30
Author(s):  
Aboubacar Marcos ◽  
Ambroise Soglo
Keyword(s):  
Existence Of Solutions ◽  
Broad Class ◽  
Kinetic Equations ◽  
Porous Medium Equation ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Splitting Technique ◽  
Medium Equation ◽  
Wasserstein Space

We use the steepest descent method in an Orlicz–Wasserstein space to study the existence of solutions for a very broad class of kinetic equations, which include the Boltzmann equation, the Vlasov–Poisson equation, the porous medium equation, and the parabolic p-Laplacian equation, among others. We combine a splitting technique along with an iterative variational scheme to build a discrete solution which converges to a weak solution of our problem.

scholarly journals The Strong Convergence of Prediction-Correction and Relaxed Hybrid Steepest-Descent Method for Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Haiwen Xu
Keyword(s):  
Variational Inequalities ◽  
Strong Convergence ◽  
Numerical Experiments ◽  
Descent Method ◽  
Steepest Descent ◽  
Steepest Descent Method ◽  
Hybrid Steepest Descent Method ◽  
Descent Directions

We establish the strong convergence of prediction-correction and relaxed hybrid steepest-descent method (PRH method) for variational inequalities under some suitable conditions that simplify the proof. And it is to be noted that the proof is different from the previous results and also is not similar to the previous results. More importantly, we design a set of practical numerical experiments. The results demonstrate that the PRH method under some descent directions is more slightly efficient than that of the modified and relaxed hybrid steepest-descent method, and the PRH Method under some new conditions is more efficient than that under some old conditions.

Steepest descent integration: A novel method for computing wavefields radiated from borehole sources

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. D151-D164
Author(s):  
Yihe Xu ◽  
Baoshan Wang ◽  
Tao Xu
Keyword(s):  
Real Axis ◽  
Steepest Descent ◽  
Oscillatory Integrals ◽  
The Real ◽  
Air Gun ◽  
Highly Oscillatory ◽  
Novel Method ◽  
Highly Oscillatory Integrals ◽  
Steepest Descent Path ◽  
Receiver Distance

Borehole sources, including chemical explosives, air gun, water gun, and piezoelectric transducers in the borehole, generate seismic waves inside and outside the borehole. Modeling the wavefield is of key importance in acoustic logging, crosshole tomography, mining geophysics, and deep sounding seismic for interpretation of amplitude information of real data and prediction of energy-radiation patterns. Classic methods for modeling the wavefield inside a borehole, such as real-axis integration, are challenged by highly oscillatory integrals encountered when modeling the wavefield outside the borehole. We have developed a novel method, called steepest descent integration (SDI), which evaluates the oscillatory wavenumber integration by numerically integrating along the steepest descent path. The oscillation along the new integration path is significantly reduced. The contributions of poles and branch cuts are added if they are located between the steepest descent path and the real axis. The SDI is applicable to arbitrary frequency and source-receiver distance. Comparison with real-axis integration shows that the method can compute highly oscillatory integrals with better efficiency and accuracy. In addition, the SDI is more numerically robust because it generates no spurious arrivals, which are evident in the real-axis integration. Analysis of numerical examples at different source-receiver distance shows that SDI is more efficient when computing far-field seismograms. This SDI can also be used to compute highly oscillatory integral in other wave-propagation problems.

The removal and morphology of intact biofilms from implanted venous access devices

1995 ◽  
Vol 53 ◽  
pp. 1020-1021
Author(s):  
M.A. Gregory ◽  
G.P. Hadley
Keyword(s):  
Patient Selection ◽  
Surgical Oncology ◽  
Venous Access ◽  
Common Procedure ◽  
Indwelling Catheters ◽  
Careful Patient ◽  
Novel Method ◽  
The Subject ◽  
Vascular Catheters ◽  
Venous Access Devices

The insertion of implanted venous access systems for children undergoing prolonged courses of chemotherapy has become a common procedure in pediatric surgical oncology. While not permanently implanted, the devices are expected to remain functional until cure of the primary disease is assured. Despite careful patient selection and standardised insertion and access techniques, some devices fail. The most commonly encountered problems are colonisation of the device with bacteria and catheter occlusion. Both of these difficulties relate to the development of a biofilm within the port and catheter. The morphology and evolution of biofilms in indwelling vascular catheters is the subject of ongoing investigation. To date, however, such investigations have been confined to the examination of fragments of biofilm scraped or sonicated from sections of catheter. This report describes a novel method for the extraction of intact biofilms from indwelling catheters.15 children with Wilm’s tumour and who had received venous implants were studied. Catheters were removed because of infection (n=6) or electively at the end of chemotherapy.

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