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Parametric Mesh Generation for Optimization
Published in S. Ratnajeevan H. Hoole, Yovahn Yesuraiyan R. Hoole, Finite Elements-based Optimization, 2019
S. Ratnajeevan H. Hoole, Yovahn Yesuraiyan R. Hoole
A Dirichlet boundary condition means the potential along the given boundary is fixed and a Neumann boundary condition means the derivative of the potential along the given boundary is fixed, and usually zero. Dirichlet boundary conditions can be implemented by keeping the potential of all the points on the given boundary to be fixed at their given value. The user can select any segment and define the potential of that segment. If the potential of the segment is set, then all the points which will be added onto that line will get this potential automatically. The boundaries that do not implement Dirichlet conditions will automatically act as Neumann boundaries during the FEA process. This is because it is natural to the finite element formulation (Hoole, 1988). Therefore, no special provisions are needed to define Neumann boundaries.
Computational Fluid Dynamics and Multiphysics
Published in Mariano Martín Martín, Introduction to Software for Chemical Engineers, 2019
Bostjan Hari, Borja Hernández, Mariano Martín
Every partial differential equation needs an initial value or guess for numerical solver to start computing the equations. On the other hand, boundary conditions are specific for each conservation equation, described in Section 8.2. The variable in the continuity equation and momentum equations is the velocity vector, the variable in the energy equation is the temperature vector and the variable in the species equation is the concentration vector. Therefore, appropriate velocity, temperature and concentration values, that represent real-world values, need to be prescribed on each computational boundary, such as inlet, outlet or wall at time zero. The prescribed values on boundaries are called boundary conditions. Each boundary condition needs to be prescribed on a node or line for two-dimensional system or on a plane for three-dimensional system. In general, there are several types of boundary conditions where the Dirichlet and Neumann are the most widely used in CFD and multiphysics applications. The Dirichlet boundary condition specifies the value on a specific boundary, such as velocity, temperature or concentration. On the contrary, the Neumann boundary condition specifies the derivative on a specific boundary, such as heat flux or diffusion flux. Once the appropriate boundary conditions are prescribed to all boundaries on the two-dimensional or three-dimensional model, the set of the conservation equations is closed and the computational model can be executed.
Fluid Flow and Its Modeling Using Computational Fluid Dynamics
Published in Shyam S. Sablani, M. Shafiur Rahman, Ashim K. Datta, Arun S. Mujumdar, Handbook of Food and Bioprocess Modeling Techniques, 2006
The solution of the governing equations requires appropriate boundary conditions and initial conditions for the fluid domain. The equations are valid for most flows, and their unique solution depends on the specification of flow conditions at the domain boundaries. The boundary conditions can be thought of as operating conditions. Depending on the problem, either a degree of freedom can be constrained (e.g., defined velocity component or pressure), or surface forces or fluxes can be applied (e.g., mass flux) on any boundary. The former is also referred to as Dirichlet boundary conditions and the latter as Neumann boundary condition. A third type of boundary condition is called Robin boundary condition where a linear combination of the solution and its normal derivative is specified.
An implicit numerical model for solving free-surface seepage problems
Published in ISH Journal of Hydraulic Engineering, 2022
The developed model includes Dirichlet boundary condition (upstream and downstream), Neumann boundary condition (impermeable surfaces), and seepage face boundary condition (free seepage face). In the presented model, domain nodes are regarded as inner nodes in the main computational algorithm, and boundary nodes are considered as bounds. Prior to proceeding to the next time step, all boundary values are updated according to conditions given at that time level. For Dirichlet boundary conditions, the fixed value is imposed on the boundary node at each time step. In the Neumann boundary condition, the control volume of the boundary vertex in the mesh vertex layout has a different shape from the inner ones. In this case, the fluxes passing through each boundary line are calculated as the fluxes passing through the other faces of that control volume, yet the boundary value divides into two components that contribute to two endpoints of the line in the control volume. More information about this approach can be found in Namin et al. (2004). For the seepage face boundary condition, neither piezometric head nor passing flux is defined. This boundary condition is imposed by making pore pressure on such a face zero.
Physics-based simulation ontology: an ontology to support modelling and reuse of data for physics-based simulation
Published in Journal of Engineering Design, 2019
Hyunmin Cheong, Adrian Butscher
The two main types are known as Dirichlet and Neumann boundary conditions, in the order presented above. These conditions are applied on the subsets of the boundary surface of the body, denoted as and , respectively. The Dirichlet boundary condition specifies that the temperature at is equal to some prescribed temperature, h. The Neumann boundary condition specifies that , the spatial rate of change of temperature in the normal direction to , is proportional to some prescribed temperature flux, g. This models conductive heat transfer across the surface. One could also have a Robin boundary condition, which is a combination of Dirichlet and Neumann boundary conditions, that can be used to describe temperature-dependent surface fluxes for example.