Vector spherical harmonics and their application to classical electrodynamics

Carrascal, B; Estévez, Gentil A.; Lee, Peilian; Lorenzo, V

doi:10.1088/0143-0807/12/4/007

Cited by 29 publications

(25 citation statements)

References 4 publications

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“…To first order, the resulting flow is given by Lamb's solutions for flow around a sphere, which can be expressed as vector spherical harmonics [46]. Solving the Stokes equation with flow and stress continuity conditions at the droplet interface gives expressions for d ( ) v lm i (as defined in [34] and supplementary information appendix A) in terms of dc lm and dR lm . The perturbation on the interface is also coupled to a perturbation of the internal concentration ρ such that å å We obtain analytical solutions for the stability by assuming a quasistatic solution for dr (taking r = 0).…”

Section: Linear Stability Analysismentioning

confidence: 99%

Instabilities, motion and deformation of active fluid droplets

Whitfield

Hawkins

2016

New J. Phys.

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We consider two minimal models of active fluid droplets that exhibit complex dynamics including steady motion, deformation, rotation and oscillating motion. First we consider a droplet with a concentration of active contractile matter adsorbed to its boundary. We analytically predict activity driven instabilities in the concentration profile, and compare them to the dynamics we find from simulations. Secondly, we consider a droplet of active polar fluid of constant concentration. In this system we predict, motion and deformation of the droplets in certain activity ranges due to instabilities in the polarisation field. Both these systems show spontaneous transitions to motility and deformation which resemble dynamics of the cell cytoskeleton in animal cells.

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Section: Linear Stability Analysismentioning

confidence: 99%

Instabilities, motion and deformation of active fluid droplets

Whitfield

Hawkins

2016

New J. Phys.

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“…21 shows DoF-refinement convergence plots for the PDE-constrained and DG-based coupling approaches: we can still guess algebraic convergence. The geometry represents Ω f , Ω 1 m , and Ω 2 m .…”

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confidence: 97%

Coupling finite elements and auxiliary sources

Casati¹,

Hiptmair²

2019

Computers & Mathematics with Applications

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We propose a hybrid method to solve time-harmonic Maxwell's equations in R 3 through the Finite Element Method (FEM) in a bounded region encompassing parameter inhomogeneities, coupled with the Multiple Multipole Program (MMP) in the unbounded complement.FEM and MMP enjoy complementary capabilities. FEM requires a mesh of the computational domain of interest. This is expensive, but can treat inhomogeneous materials, shapes with corners, or other singularities. Moreover, FEM allows a purely local construction of the discrete system of equations.On the other hand, MMP belongs to the class of methods of auxiliary sources and of Trefftz methods, as it employs point sources that are exact solutions of the homogeneous equations as (global) basis functions: one only needs to evaluate them on hypersurfaces to obtain the discrete problem, and the resulting linear combination is valid in the whole domain where the equations hold, which can be unbounded. MMP performs well where the electromagnetic field is smooth, i.e. in the free space far from physical sources and material interfaces.Thus, a natural way to combine the strengths of these methods arises when one needs to simulate the electromagnetic field of components with inhomogeneous parameters or nonsmooth shapes surrounded by free space: use FEM on a mesh defined on the components and MMP in the unbounded domain outside. The boundary between the FEM and MMP domains can be nonphysical if one surrounds the components by a conforming mesh of an "airbox" also modeled by FEM.The interface conditions on the surface of the FEM domain are key to accurate coupled FEM-MMP solutions. Integrating by parts the variational form solved by FEM, surface integrals appear, through which one can impose interface conditions by substituting the ansatz of MMP.iii However, one interface condition (for example, the continuity of the tangential trace for Maxwell's equations) cannot be imposed in this way.To include this additional condition, we derive four methods from Lagrangian functionals that enforce both the variational form in the FEM domain and all the interface conditions between different discretizations:1. Least-squares-based coupling using techniques from PDE-constrained optimization. This approach optimizes a functional for the additional condition, subject to a constraint expressed by the variational form of FEM.2. Discontinuous Galerkin (DG) between the meshed FEM domain and the single-entity "MMP mesh", interpreting MMP as FEM with special basis functions.3. Multi-field variational formulation in the spirit of mortar finite element methods, relying on the same interpretation of MMP as the DG-based coupling.4. Coupling through the Dirichlet-to-Neumann operator : here we introduce a weak formulation of the additional condition where MMP basis functions are used as test functions. This approach is generalized to couple MMP with any numerical method based on volume meshes that fits co-chain calculus.Furthermore, to minimize the meshed region and for the sake of generalization, we as...

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“…On the other hand, the spherical dependence g is usually formulated in terms of vector spherical harmonics, 24 p. 289. The additional constraint of the Coulomb gauge in (1a) is taken into account by selecting a subset of vector spherical harmonics to express g .…”

Section: Trefftz Methodsmentioning

confidence: 99%

Coupling finite elements and auxiliary sources for electromagnetic wave propagation

Casati

Hiptmair

Smajić

2020

Int J Numerical Modelling

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We propose four approaches to solve time-harmonic Maxwell's equations in R 3 through the finite element method (FEM) in a bounded region encompassing parameter inhomogeneities, coupled with the multiple multipole program (MMP) in the unbounded complement. MMP belongs to the class of methods of auxiliary sources and of Trefftz methods, as it employs point sources that spawn exact solutions of the homogeneous equations. Each of these sources is anchored at a point that, if singular, is placed outside the respective domain of approximation. In the MMP domain, we assume that material parameters are piecewise constant, which induces a partition into one unbounded subdomain and other bounded, but possibly very large, subdomains, each requiring its own MMP trial space. Hence, in addition to the transmission conditions between the FEM and MMP domains, one also has to impose transmission conditions connecting the MMP subdomains. Coupling approaches arise from seeking stationary points of Lagrangian functionals that both enforce the variational form of the equations in the FEM domain and match the different trial functions across subdomain interfaces. We discuss the following approaches: 1 Least-squares-based coupling using techniques from PDE-constrained optimization. 2 Discontinuous Galerkin coupling between the meshed FEM domain and the single-entity MMP subdomains. 3 Multi-field variational formulation in the spirit of mortar FEMs. 4 Coupling through the Dirichlet-to-Neumann operator. We compare these approaches in a series of numerical experiments with different geometries and material parameters, including examples that exhibit triple-point singularities and infinite layered media.

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Vector spherical harmonics and their application to classical electrodynamics

Cited by 29 publications

References 4 publications

Instabilities, motion and deformation of active fluid droplets

Instabilities, motion and deformation of active fluid droplets

Coupling finite elements and auxiliary sources

Coupling finite elements and auxiliary sources for electromagnetic wave propagation

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