Advances in Mathematical Physics

Volume 2015 (2015), Article ID 469310, 12 pages

http://dx.doi.org/10.1155/2015/469310

## Comparison of 3D Adaptive Remeshing Strategies for Finite Element Simulations of Electromagnetic Heating of Gold Nanoparticles

Group for Automatic Mesh Generation and Advanced Methods, Gamma3, University of Technology of Troyes (UTT), French National Institute for Research in Computer Science and Automation (INRIA), 12 rue Marie Curie, CS 42060, 10004 Troyes Cedex, France

Received 25 February 2015; Accepted 29 April 2015

Academic Editor: Ricardo Weder

Copyright © 2015 Fadhil Mezghani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The optical properties of metallic nanoparticles are well known, but the study of their thermal behavior is in its infancy. However the local heating of surrounding medium, induced by illuminated nanostructures, opens the way to new sensors and devices. Consequently the accurate calculation of the electromagnetically induced heating of nanostructures is of interest. The proposed multiphysics problem cannot be directly solved with the classical refinement method of Comsol Multiphysics and a 3D adaptive remeshing process based on an *a posteriori* error estimator is used. In this paper the efficiency of three remeshing strategies for solving the multiphysics problem is compared. The first strategy uses independent remeshing for each physical quantity to reach a given accuracy. The second strategy only controls the accuracy on temperature. The third strategy uses a linear combination of the two normalized targets (the electric field intensity and the temperature). The analysis of the performance of each strategy is based on the convergence of the remeshing process in terms of number of elements. The efficiency of each strategy is also characterized by the number of computation iterations, the number of elements, the CPU time, and the RAM required to achieve a given target accuracy.

#### 1. Introduction

Noble metals nanoparticles, particularly gold nanoparticles, are known as efficient light absorbers [1–3]. Their fascinating optical properties have been studied intensively. Recently, the heat production by these nanoparticles under optical illumination has also aroused much interest [4]. Indeed, an optically excited nanoparticle absorbs a part of the electromagnetic energy that can heat not only the particle itself but also its surrounding [5]. This heating effect becomes particularly strong if the incident electromagnetic wave frequency is near the plasmon resonance frequency of the nanoparticle [6, 7]. The increase of temperature in gold nanoparticles has a variety of applications in nanotechnology, biology, chemistry, and medicine (such as the photothermal cancer therapy using thermal necrosis of tumor cells) [4, 8] and the drug delivery with the remote release of drugs from a capsule containing gold nanoparticles when excited by a laser source [9–11].

Owing to these applications, accurate numerical studies of the heating of gold nanoparticle induced by a laser source become pertinent. The problem consists in solving two partial differential equations (PDEs) corresponding to both physics, the result of the electromagnetic problem being the source of heat.

The finite element method is widely used for the solution of the partial differential equations, particularly the Helmholtz equation [12] and the heat equation [13]. The mesh quality in the finite element method allows the control of the accuracy of the numerical solutions and the computational memory [14, 15]. The adaption is a powerful technique that can produce an optimal mesh for the finite element calculation [15, 16]. The remeshing methods that produce a new mesh at each iteration of the remeshing loop distinguish themselves from the refinement methods which overall maintain the old mesh but either move some amount of nodes (-method), increase the degree of interpolation polynomials (-method), or increase the mesh density by inserting nodes (-method) or perform a combination of the mentioned methods. The objective of the mesh coarsening is to meet the accuracy requirements fixed by an error estimator that can be physical or geometrical [17, 18]. In this paper, a -adaptive remeshing process uses a physical error estimator based on an* a posteriori* p1-interpolation of the physical solution to increase the accuracy of the solution while ensuring the convergence of calculation. Such an* a posteriori* estimator uses the interpolation of the physical solution computed at the previous step to construct a new mesh that respects the geometry of objects defined in the physical problem. The optimization of the mesh is governed by the Hessian of the interpolation of the solution at the previous step of remeshing. Let us note that the classical -method inserting nodes on elements edges and therefore dividing each cell of the initial mesh, keeping constant the position of nodes at the previous step of refinement, is not able to solve the investigated problem.

Three strategies governing the remeshing are proposed. Each strategy differs from the others by the physical solution that is used for the remeshing process. The accuracy is targeted either on both the electric field and the temperature, on the temperature, or on a linear combination of both, leading to different behavior of the algorithms.

Section 2 presents the description of the photothermal model and its formulation for finite element method. Section 3 is devoted to both the remeshing adaptive method and the strategies of remeshing using three different choices of physical solutions. In Section 4 numerical results obtained from the three strategies of remeshing are discussed and compared before concluding.

#### 2. Physics: Electromagnetic and Thermic Problems

In this section we consider a gold nanoparticle of relative permittivity (complex number) and thermal conductivity . The gold nanoparticle is immersed in a uniform dielectric medium with a relative permittivity and a thermal conductivity (Figure 1). Both media are considered linear, homogeneous, nonmagnetic, and isotropic. The external medium is supposed to be nonabsorbing; therefore the imaginary part of the relative permittivity vanishes: .