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In this work we test the numerical behaviour of matrix-valued fields approximated by finite element subspaces of [H1]3×3, [H(curl)]3 and H(symCurl) for a linear abstract variational problem connected to the relaxed micromorphic model. The formulation of the corresponding finite elements is introduced, followed by numerical benchmarks and our conclusions. The relaxed micromorphic continuum model reduces the continuity assumptions of the classical micromorphic model by replacing the full gradient of the microdistortion in the free energy functional with the Curl. This results in a larger solution space for the microdistortion, namely [H(curl)]3 in place of the classical [H1]3×3. The continuity conditions on the microdistortion can be further weakened by taking only the symmetric part of the Curl. As shown in recent works, the new appropriate space for the microdistortion is then H(symCurl). The newly introduced space gives rise to a new differential complex for the relaxed micromorphic continuum theory., Journal of engineering mathematics;136(1)

One approach for the simulation of metamaterials is to extend an associated continuum theory concerning its kinematic equations, and the relaxed micromorphic continuum represents such a model. It incorporates the Curl of the nonsymmetric microdistortion in the free energy function. This suggests the existence of solutions not belonging to H1, such that standard nodal H1-finite elements yield unsatisfactory convergence rates and might be incapable of finding the exact solution. Our approach is to use base functions stemming from both Hilbert spaces H1 and H(curl), demonstrating the central role of such combinations for this class of problems. For simplicity, a reduced two-dimensional relaxed micromorphic continuum describing antiplane shear is introduced, preserving the main computational traits of the three-dimensional version. This model is then used for the formulation and a multi step investigation of a viable finite element solution, encompassing examinations of existence and uniqueness of both standard and mixed formulations and their respective convergence rates., Computational mechanics;Vol. 68. 2021, pp 1���24

Elektromagnetski problemi kao što je raspršenje elektromagnetskog vala, osim u slučajevima visoke simetrije, zadaju se na složenim geometrijama i nisu analitički rješivi. Numeričke metode savladavaju navedene probleme, a sa složenim geometrijama najbolje se nosi metoda konačnih elemenata. Zbog vektorske prirode problema, standardni konačni elementi nisu dobri te numerička rješenja dobivena standardnim, po dijelovima neprekinutim polinomijalnim elementima, ne konvergiraju prema stvarnom rješenju problema. Za dobivanje stabilnog numeričkog rješenja potrebno je koristiti rubne konačne elemente. Nedelecovi konačni elementi su primjer rubnih konačnih elemenata pogodnih za rješavanje vektorskih problema koji u sebi sadrže vektorski operator rotacije. U okviru ovog rada dan je pregled metode konačnih elemenata primijenjene na problem raspršenja elektromagnetskog vala. Izvedena je slaba formulacija problema i dana je referentna implementacija Nedelecovih konačnih elemenata. U radu su opisane optimizacije u vidu smanjenja broja operacija s pomičnim zarezom. U konačnici je postavljanje sustava vremenski uspoređeno s programskom bibliotekom FEniCS. Za matricu sa 60 004 000 elemenata koji nisu nula, sustav jednadžbi je generiran dvostruko brže koristeći referentnu implementaciju opisanu u okviru ovog rada. Electromagnetic problems such as scattering of electromagnetic waves can not be solved analytically except for highly symmetric cases in linear materials. Modern ways of dealing with that problem are by solving it numerically. Complex geometries are best dealt with using finite element method. Because of vector nature of the problem, standard piecewise polynomial finite elements can yield spurious solutions. On the other hand, edge elements provide numerically stable solutions. Example of such elements are Nedelec finite elements which are found to be appropriate to handle curl operator. In this thesis, a brief introduction to the finite element method is given on the example of scattering of electromagnetic waves. Weak formulation of the problem is derived and reference implementation of Nedelec finite elements is given. Optimization techniques in terms of reducing the floating-point operation count are given. Matrix assembly was benchmarked against well-known finite element framework, FEniCS. Reference implementation presented in this thesis assembles the finite element matrix with 60 004 000 non-zero entries twice as fast as the FEniCS framework.

Humanitarian clearance of minefields is most often carried out by hand, conventionally using a a metal detector and a probe. Detection is a very slow process, as every piece of detected metal must treated as if it were a landmine and carefully probed and excavated, while many of them are not. The process can be safely sped up by use of Ground-Penetrating Radar (GPR) to image the subsurface, to verify metal detection results and safely ignore any objects which could not possibly be a landmine. In this thesis, we explore the possibility of using Full Wave Inversion (FWI) to improve GPR imaging for landmine detection. Posing the imaging task as FWI means solving the large-scale, non-linear and ill-posed optimisation problem of determining the physical parameters of the subsurface (such as electrical permittivity) which would best reproduce the data. This thesis begins by giving an overview of all the mathematical and implementational aspects of FWI, so as to provide an informative text for both mathematicians (perhaps already familiar with other inverse problems) wanting to contribute to the mine detection problem, as well as a wider engineering audience (perhaps already working on GPR or mine detection) interested in the mathematical study of inverse problems and FWI.We present the first numerical 3D FWI results for GPR, and consider only surface measurements from small-scale arrays as these are suitable for our application. The FWI problem requires an accurate forward model to simulate GPR data, for which we use a hybrid finite-element boundary-integral solver utilising first order curl-conforming N\'{e}d\'{e}lec (edge) elements. We present a novel `line search' type algorithm which prioritises inversion of some target parameters in a region of interest (ROI), with the update outside of the area defined implicitly as a function of the target parameters. This is particularly applicable to the mine detection problem, in which we wish to know more about some detected metallic objects, but are not interested in the surrounding medium. We may need to resolve the surrounding area though, in order to account for the target being obscured and multiple scattering in a highly cluttered subsurface.We focus particularly on spatial sensitivity of the inverse problem, using both a singular value decomposition to analyse the Jacobian matrix, as well as an asymptotic expansion involving polarization tensors describing the perturbation of electric field due to small objects. The latter allows us to extend the current theory of sensitivity in for acoustic FWI, based on the Born approximation, to better understand how polarization plays a role in the 3D electromagnetic inverse problem. Based on this asymptotic approximation, we derive a novel approximation to the diagonals of the Hessian matrix which can be used to pre-condition the GPR FWI problem. Humanitarian clearance of minefields is most often carried out by hand, conventionally using a a metal detector and a probe. Detection is a very slow process, as every piece of detected metal must treated as if it were a landmine and carefully probed and excavated, while many of them are not. The process can be safely sped up by use of Ground-Penetrating Radar (GPR) to image the subsurface, to verify metal detection results and safely ignore any objects which could not possibly be a landmine. In this thesis, we explore the possibility of using Full Wave Inversion (FWI) to improve GPR imaging for landmine detection. Posing the imaging task as FWI means solving the large-scale, non-linear and ill-posed optimisation problem of determining the physical parameters of the subsurface (such as electrical permittivity) which would best reproduce the data. We consider the mathematical and implementational aspects of such an approach, presenting the first FWI results for GPR in 3D. MATLAB code for simulating GPR data, solving the inverse (FWI) problem, and approximating scattered fields with polarization tensors. DOI 10.5281/zenodo.31437

In this note, we provide a conveniently implementable basis for simplicial Nédélec spaces of any order in any space dimension. The main feature of the basis is that it is expressed solely in terms of the barycentric coordinates of the simplex.

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