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Conclusion
Published in Pavel Sumets, Computational Framework for the Finite Element Method in MATLAB® and Python, 2023
In this book we have considered creating computational framework for the linear finite element method (FEM) with application to solving ordinary and partial differential equations having mixed boundary conditions. Each step of the FEM algorithm is described in detail and programming codes in MATLAB and Python are composed. Although there exists various software packages implementing FEM, understanding the basics of the FEM algorithm is important to utilize these packages. Explanations of the FEM algorithm and computational framework given in this book not only help to understand basics of the method but to provide guidance of how to build customized code for solving a problem using FEM. The latter is important since it allows to develop coding skills and to become more flexible with solving any nonstandard problem where standard FEM packages could fail to solve (or simply these packages are not available). Apart from this, coding functions from this book can be extended or modified to build improved FEM framework with some specific features if required.
Numerical Solutions
Published in Sadık Kakaç, Yaman Yener, Carolina P. Naveira-Cotta, Heat Conduction, 2018
Sadık Kakaç, Yaman Yener, Carolina P. Naveira-Cotta
In the preceding chapters, various analytical methods were discussed for the solution of heat conduction problems involving relatively simple geometric shapes with certain straightforward boundary conditions in the rectangular, cylindrical, and spherical coordinate systems. The vast majority of problems encountered in practice, however, cannot be solved analytically as they usually involve irregular geometries with mathematically inconvenient mixed boundary conditions. In such cases, numerical, graphical, and/or hybrid numerical–analytical methods (as will be presented in Chapter 13, and for further details, see Ref. [21]) often provide the answer. For example, an exact analytical solution for the temperature distribution in a turbine blade cannot be obtained because the boundary of the blade is not parallel to the coordinate surfaces of an orthogonal system. The geometry, however, can be simplified for an analytical solution, but the results may not be accurate enough for practical applications.
Numerical Solutions
Published in Yaman Yener, Sadık Kakaç, Heat Conduction, 2018
In the preceding chapters, various analytical methods were discussed for the solution of heat conduction problems involving relatively simple geometric shapes with certain straightforward boundary conditions in the rectangular, cylindrical, and spherical coordinate systems. The vast majority of problems encountered in practice, however, cannot be solved analytically as they usually involve irregular geometries with mathematically inconvenient mixed boundary conditions. In such cases, numerical and/or graphical methods often provide the answer. For example, an exact analytical solution for the temperature distribution in a turbine blade cannot be obtained because the boundary of the blade is not parallel to the coordinate surfaces of an orthogonal system. The geometry, however, can be simplified for an analytical solution, but the results may not be accurate enough for practical applications.
Utilising physics-informed neural networks for optimisation of diffusion coefficients in pseudo-binary diffusion couples
Published in Philosophical Magazine, 2023
Hemanth Kumar, Neelamegan Esakkiraja, Anuj Dash, Aloke Paul, Saswata Bhattacharyya
For the special case in which only two components develop the diffusion profile in the interdiffusion zone, such as in CB and PB diffusion couple experiments, the PDE-constrained optimisation procedure is formulated as: where is the composition field of the independent diffusing species, is the interdiffusion coefficient, and are the intrinsic diffusion coefficients at the K-plane. Note that there is no restriction on the type of boundary condition used – one can use Dirichlet, Neumann, periodic, or mixed boundary conditions. In the case of CB, we consider the composition variables and , while in the case of PB, we use the modified composition variables, and . The contributions to the composite loss function are: Here, and . Thus,
A theoretical study on ground surface settlement induced by a braced deep excavation
Published in European Journal of Environmental and Civil Engineering, 2022
Haohua Chen, Jingpei Li, Changyi Yang, Ce Feng
This paper conducts a theoretical study on the deep excavation induced GSS by taking both the wall deformation and the stress release into consideration. Based on the elastic theory, the problem considered is formulated as a system of second-order partial differential equations with mixed boundary conditions. According to the superposition principle, the mixed boundary conditions are decomposed into displacement and stress boundary conditions, which are solved by the Separation of Variable Method (SVM) and the Fourier Transform Method (FTM), respectively. Whereas the unknown coefficients of the wall deflection function (displacement boundary) are determined by a novel least square method. The validity of the proposed theoretical approach is checked by predicting the GSS from several published cases in Shanghai soft clay. The good agreements demonstrate the capability of the proposed method in predicting the GSS induced by deep excavation in soft soils.
Discontinuous Galerkin isogeometric analysis for segmentations generating overlapping regions
Published in Applicable Analysis, 2021
Christoph Hofer, Ioannis Toulopoulos
The weak formulation of the boundary value problem (1) reads as follows: for given source function find a function such that the variational identity is satisfied, where the bilinear form and the linear form are defined by respectively. The given diffusion coefficient is assumed to be uniformly positive and piece-wise (patch-wise, see below) constant. These assumptions ensure the existence and uniqueness of the solution due to Lax-Milgram's lemma. For simplicity, we only consider pure Dirichlet boundary conditions on . However, the analysis presented in our paper can easily be generalized to other constellations of boundary conditions that ensure existence and uniqueness such as Robin or mixed boundary conditions.