Analysis of Shape Optimization for Magnetic Microswimmers

Walker, Shawn W.; Keaveny, Eric E.

doi:10.1137/110845823

Cited by 5 publications

(8 citation statements)

References 71 publications

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“…The upper-level optimization problem is solved by a gradient descent scheme posed in function space. See [71,75] for related descent schemes.…”

Section: Introductionmentioning

confidence: 99%

Droplet Footprint Control

Laurain¹,

Walker²

2015

SIAM J. Control Optim.

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Controlling droplet shape via surface tension has numerous technological applications, such as droplet lenses and lab-on-a-chip. This motivates a partial differential equationconstrained shape optimization approach for controlling the shape of droplets on flat substrates by controlling the surface tension of the substrate. We use shape differential calculus to derive an L 2 gradient flow approach to compute equilibrium shapes for sessile droplets on substrates. We then develop a gradient-based optimization method to find the substrate surface tension coefficient yielding an equilibrium droplet shape with a desired footprint (i.e., the liquid-solid interface has a desired shape). Moreover, we prove a sensitivity result with respect to the substrate surface tensions for the free boundary problem associated with the footprint. Numerical results are also presented to showcase the method.

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“…The upper-level optimization problem is solved by a gradient descent scheme posed in function space. See [71,75] for related descent schemes.…”

Section: Introductionmentioning

confidence: 99%

Droplet Footprint Control

Laurain¹,

Walker²

2015

SIAM J. Control Optim.

Self Cite

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“…For further details on SQP methods, see [10,21,38,25,51]. Similar approaches can be found in [58,57,27]. Update curve:…”

Section: Sqp Methodsmentioning

confidence: 99%

“…for all smooth functions V and L 2 (Σ) functions p, where p is a Lagrange multiplier (see [57] for a similar approach). We use δ Σ to emphasize that we are perturbing with respect to Σ.…”

Section: Local Inextensibility Constraintmentioning

confidence: 99%

“…In [20,42], they showed the existence of C 1,1 minimizing curves that satisfied a thickness constraint. A similar problem was also analyzed for optimizing the shape of magnetic micro-swimmers, with a local length constraint, in [27,57]. We will not delve into the theoretical intricacies of the continuous optimization problem (50), because most of the theoretical issues have been established in other research.…”

Section: Minimization Problemmentioning

confidence: 99%

“…Evaluate θ(Σ k h , ζ k+1 ) and θ(Σ k+1 h , ζ k+1 ) by (57), and δθ(Σ k h , ζ k+1 ; V k+1 h ) by (60).…”

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confidence: 99%

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Shape optimization of self-avoiding curves

Walker

2016

Journal of Computational Physics

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This paper presents a softened notion of proximity (or self-avoidance) for curves. We then derive a sensitivity result, based on shape differential calculus, for the proximity. This is combined with a gradient-based optimization approach to compute three-dimensional, parameterized curves that minimize the sum of an elastic (bending) energy and a proximity energy that maintains self-avoidance by a penalization technique. Minimizers are computed by a sequential-quadratic-programming (SQP) method where the bending energy and proximity energy are approximated by a finite element method. We then apply this method to two problems. First, we simulate adsorbed polymer strands that are constrained to be bound to a surface and be (locally) inextensible. This is a basic model of semi-flexible polymers adsorbed onto a surface (a current topic in material science). Several examples of minimizing curve shapes on a variety of surfaces are shown. An advantage of the method is that it can be much faster than using molecular dynamics for simulating polymer strands on surfaces. Second, we apply our proximity penalization to the computation of ideal knots. We present a heuristic scheme, utilizing the SQP method above, for minimizing rope-length and apply it in the case of the trefoil knot. Applications of this method could be for generating good initial guesses to a more accurate (but expensive) knot-tightening algorithm.

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Simulating squirmers with volumetric solvers

Paz

Buscaglia

2020

J Braz. Soc. Mech. Sci. Eng.

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Squirmers are models of a class of microswimmers that self-propel in fluids without significant deformation of their body shape, such as ciliated organisms and phoretic particles. Available techniques for their simulation are based on the boundary element method and do not contemplate nonlinearities such as those arising from the inertia of the fluid or non-Newtonian rheology. This article describes a methodology to simulate squirmers that overcomes these limitations by using volumetric numerical methods, such as finite elements or finite volumes. It deals with interface conditions at the surface of the squirmer that generalize those in the published literature, which are generally restricted to the imposition of slip velocities. The actual procedures to be performed on a fluid solver to implement the proposed methodology are provided, including the treatment of metachronal surface waves. Among the several numerical examples, a two-dimensional simulation is shown of the hydrodynamic interaction of two individuals of Opalina ranarum.

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Analysis of Shape Optimization for Magnetic Microswimmers

Cited by 5 publications

References 71 publications

Droplet Footprint Control

Droplet Footprint Control

Shape optimization of self-avoiding curves

Simulating squirmers with volumetric solvers

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