Mixed enrichment for the finite element method in heterogeneous media

Problems of multiple scales of interest or of locally nonsmooth solutions may often involve heterogeneous media. These problems are usually very demanding in terms of computations with the conventional finite element method. On the other hand, different enriched finite element methods such as the partition of unity, which proved to be very successful in treating similar problems, are developed and studied for homogeneous media. In this work, we present a new idea to extend the partition of unity finite element method to treat heterogeneous materials. The idea is studied in applications to wave scattering and heat transfer problems where significant advantages are noted over the standard finite element method. Although presented within the partition of unity context, the same enrichment idea can also be extended to other enriched methods to deal with heterogeneous materials

Iterative solvers for generalized finite element solution of boundary-value problems

Most of generalized finite element methods use dense direct solvers for the resulting linear systems. This is mainly the case due to the ill‐conditioned linear systems that are associated with these methods. In this study, we investigate the performance of a class of iterative solvers for the generalized finite element solution of time‐dependent boundary‐value problems. A fully implicit time‐stepping scheme is used for the time integration in the finite element framework. As enrichment, we consider a combination of exponential functions based on an approximation of the internal boundary layer in the problem under study. As iterative solvers, we consider the changing minimal residual method based on the Hessenberg reduction and the generalized minimal residual method. Compared with dense direct solvers, the iterative solvers achieve high accuracy and efficiency at low computational cost and less storage as only matrix–vector products are involved in their implementation. Two test examples for boundary‐value problems in two space dimensions are used to assess the performance of the iterative solvers. Comparison to dense direct solvers widely used in the framework of generalized finite element methods is also presented. The obtained results demonstrate the ability of the considered iterative solvers to capture the main solution features. It is also illustrated for the first time that this class of iterative solvers can be efficient in solving the ill‐conditioned linear systems resulting from the generalized finite element methods for time domain problems

Identifying the wavenumber for the inverse Helmholtz problem using an enriched finite element formulation

We investigate the inverse problem of identifying the wavenumber for the Helmholtz equation. The problem solution is based on measurements taken at few points from inside the computational domain or on its boundary. A novel iterative approach is proposed based on coupling the secant and the descent methods with the partition of unity method. Starting from an initial guess for the unknown wavenumber the forward problem is solved using the partition of unity method. Then the secant/descent methods are used to improve the initial guess by minimizing a predefined objective function based on the difference between the solution and a set of data points. In the next round of iterations the improved wavenumber estimate is used for the forward problem solution and the partition of unity approximation is improved by adding more enrichment functions. The iterative process is terminated when the objective function has converged and a set of two predefined tolerances are met. To evaluate the estimate accuracy we propose to utilize extra data points. To validate the approach and test its efficiency two wave applications with known analytical solutions are studied. The results show that the proposed approach can achieve high accuracy for the studied applications even when the considered data is contaminated with noise. Despite the clear advantages that were previously shown in the literature for solving the forward Helmholtz problem, this work presents a first attempt to solve the inverse Helmholtz problem with an enriched finite element approach

Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference

The 6th ECCOMAS Young Investigators Conference YIC2021 will take place from July 7th through 9th, 2021 at Universitat Politècnica de València, Spain. The main objective is to bring together in a relaxed environment young students, researchers and professors from all areas related with computational science and engineering, as in the previous YIC conferences series organized under the auspices of the European Community on Computational Methods in Applied Sciences (ECCOMAS). Participation of senior scientists sharing their knowledge and experience is thus critical for this event.YIC 2021 is organized at Universitat Politécnica de València by the Sociedad Española de Métodos Numéricos en Ingeniería (SEMNI) and the Sociedad Española de Matemática Aplicada (SEMA). It is promoted by the ECCOMAS.The main goal of the YIC 2021 conference is to provide a forum for presenting and discussing the current state-of-the-art achievements on Computational Methods and Applied Sciences,including theoretical models, numerical methods, algorithmic strategies and challenging engineering applications.Nadal Soriano, E.; Rodrigo Cardiel, C.; Martínez Casas, J. (2022). Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. https://doi.org/10.4995/YIC2021.2021.15320EDITORIA

Generalized finite elements for transient heat diffusion problems

For many decades, the classical Finite Element Method (FEM) was successfully used to solve a wide range of problems that are governed by the scalar transient diffusion equation. It produced robust solutions with remarkable accuracy for a variety of problems with complex geometries and boundary conditions. However, the numerical solution still poses a serious challenge when it diffuses with steep gradients. This situation arises in many engineering problems, such as in glass cooling, where the temperature difference between the cooling object and the ambient environment is so large that it leads to severe thermal stresses. To properly model this behaviour, the conventional FEM uses highly refined mesh grids to accommodate the sharp change in the temperature field. Given that the problem is time dependent, computing the solution over refined meshes for thousands of time steps leads to prohibitively expensive solutions. To address this limitation, this thesis aims to assess a novel approach based on time-independent field enrichment for efficiently solving time-dependent heat diffusion problems over coarse mesh grids. The approach consists to incorporate a-priori knowledge in the finite element approximation space through carefully selected functions that exhibit similar behaviour as of the true solution. In this work, Gaussian functions with various rates of decay are employed in combination with linear Lagrange polynomial-based finite elements, such that inter-element continuity is automatically satisfied. This technique provides a remarkable reduction of the computational cost, in comparison to the widely used classical low order polynomial-based FEM. To test the accuracy and reliability of this approach, computable a-posteriori residual error estimates that are mathematically rigorous; are developed and implemented for both two and three-dimensional problems. The proposed estimates are straightforward to implement and are shown to provide reliable and practical upper bounds for the numerical errors, independent of the heuristically chosen enrichment functions. The estimates accurately capture the decrease of the error as the number of enrichment functions is increased or the time step is reduced. However, ill-conditioning is shown to be an inherent feature of the field enrichment. Therefore, the proposed error estimates are used to adaptively enrich the element field in subdomains with relatively higher errors. Both the global error, in the whole space–time domain, and local error indicators in the individual elements of the mesh are investigated, for the adaptive selection of the enrichment functions. An adaptive algorithm is proposed to identify the elements with higher errors so that further enrichments are added locally; leading to significant savings in comparison to the case with uniform enrichments.James Watt scholarshi

ATOMIC SCALE SIMULATION OF ACCIDENT TOLERANT FUEL MATERIALS FOR FUTURE NUCLEAR REACTORS

The 2011 accident at the Fukushima-Daiichi power station following the earthquake and tsunami in Japan put renewed emphasis on increasing the accident tolerance of nuclear fuels. Although the main concern in this incident was the loss of coolant and the Zr cladding reacting with water to form hydrogen, the fuel element is an integral part of any accident tolerant fuel (ATF) concept. Therefore, to license a new commercial nuclear fuel, the prediction of fuel behavior during operation becomes a necessity. This requires knowledge of its properties as a function of temperature, pressure, initial fuel microstructure and irradiation history, or more precisely the changes in microstructure due to irradiation and/or oxidation. Amongst other nuclear fuels, uranium diboride (UB2) and uranium silicide (U3Si2) are considered as potential fuels for the next generation of nuclear reactors due to their high uranium density and high thermal conductivity compared to uranium dioxide (UO2). However, the thermophysical properties and behavior of these fuels under extreme conditions are not well known, neither are they readily available in the literature. Therefore, in this thesis, density functional theory (DFT) and classical molecular dynamic (MD) simulations were used to investigate the thermophysical properties, radiation tolerance and oxidation behavior of UB2 and U3Si2 as potential fuels or burnable absorbers for the next generation of nuclear reactors. UB2 was studied in order to understand its thermophysical properties as a function of temperature. The phonon-assisted thermal conductivity (kph) exhibits large directional anisotropy with larger thermal conductivity parallel to the crystal direction. This has implications for the even dissipation of heat. The increase in thermal conductivity with temperature is justified by the electronic contribution to the thermal transport, especially at high temperatures. This shows that UB2 is a potential ATF candidate. In terms of radiation tolerance, Zr is more soluble in UB2 than Xe, while uranium vacancy is the most stable solution site. Furthermore, as the concentration of Zr fission product (FP) increases, there is a contraction in the volume of UB2, while an increase in Xe results in swelling of the fuel matrix. In terms of diffusion, the presence of an FP in the neighboring U site increases the migration of U in UB2, making U migrate more readily than B as observed in the ideal system. The thermophysical properties of U3Si2 as a possible ATF were studied and discussed considering the neutronic penalty of using a SiC cladding in a reactor. The calculated molar heat capacity and experimental data are in reasonable agreement. Due to the anisotropy in lattice expansion, a directional dependence in the linear thermal expansion coefficient was noticed, which has also been experimentally observed. The thermal conductivity of U3Si2 increases with temperature due to the electronic contribution while the phonon contribution decreases with increasing temperature. A comparison of the thermal conductivity in two different crystallographic directions sheds light on the spatial anisotropy in U3Si2 fuel material. The inherent anisotropic thermophysical properties can be used to parametrize phase field models by incorporating anisotropic thermal conductivity and thermal expansion. This allows for a more accurate description of microstructural changes under variable temperature and irradiation conditions. Due to the metallic nature of U3Si2, the oxidation mechanism is of special interest and has to be investigated. Oxidation in O2 and H2O was investigated using experimental and theoretical methods. The presence of oxide signatures was established from X-ray diffraction (XRD) and Raman spectroscopy after oxidation of the solid U3Si2 sample in oxygen. Surface oxidation of U3Si2 can be linked to the significant charge transfer from surface uranium ions to water and/or oxygen molecules. Detailed charge transfer and bond length analysis revealed the preferential formation of mixed oxides of U-O and Si-O on the U3Si2 (001) surface as well as UO2 alone on the U3Si2 (110) and (111) surfaces. Formation of elongated O−O bonds (peroxo) confirmed the dissociation of molecular oxygen before U3Si2 oxidation. Experimental analysis by Raman spectroscopy and XRD of the oxidized U3Si2 samples has revealed the formation of higher uranium oxides such as UO3 and U3O8. Overall, this work serves as a step towards understanding the complex anisotropic behavior of the thermophysical properties of metallic UB2 and U3Si2 considered as potential accident tolerant nuclear fuel. The calculated anisotropy of thermophysical properties can be used to parametrize phase field model and to incorporate in it anisotropic thermal conductivity and thermal expansion