Questions tagged [boundary-conditions]

For questions regarding the choice and/or appropriateness of conditions necessary to model a particular phenomenon with a partial differential equations.

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Transport Equation in a Tube: Source Term on Boundary

I'm modeling mass transport in a flow reactor. The flow reactor is a tube, which allows me to use cylindrical symmetry in solving the Convection-Diffusion-Reaction (CDR) Equation, which governs the ...
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1answer
617 views

Problem in Discretizing Convection-Diffusion-Reaction equation

I'm trying to solve the Convection-Diffusion-Reaction (CDR) equation on a rectangular domain, using cylindrical coordinates and Finite Difference Methods (FDM) (this approximates a flow reactor). ...
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1answer
1k views

Implementation of gradient zero boundary conditon in advection-diffusion equation

My question is about Finite Element Method. I want to know how to implement "gradient zero" conditions to advection-diffusion equations in conservative form like, $\frac{\partial \rho}{\partial t} + ...
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1answer
438 views

Boundary conditions for second order PDE

For a second order PDE, for example heat conduction equation $\frac{\partial T}{\partial t} = \frac{\alpha}{C_p} \nabla^2 T$, is it possible to determine the steady-state (or even transient) solution ...
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43 views

how to find outward normal for robin codition [duplicate]

I wrote the code to fix this problem; now I should validate it using a test function to see the error in my code. I can't figure out how starting from u can derive the function g on the edge, in ...
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1answer
583 views

Pde problem with robin boundary condition

I have my pde 2D problem with robin condition (form: du/dn +ku=g) to solve with matlab. i have the exact function u and I want to find the function g in robin condition. How can i do it? thanks for ...
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2answers
161 views

How to impose a constant constraint PDE

What is the best way to impose a "constant constraint" for a PDE? Specifically, I want to solve an eigenvalue problem $Au=\lambda u$ on the rectangle $(0,2\pi)\times(-\pi/2,\pi/2)$ with periodicity ...
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1answer
410 views

The Lax-Milgram Lemma in FEM with non-homogenous Dirichlet BC

How can show that the prerequisites for the Lax-Milgram Lemma holds if I have different test and trial spaces (which I think is the natural thing to have if at least part of the boundary is non-...
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1answer
259 views

Multigrid stops converging when more grid levels are used

I'm having a problem with multigrid code I wrote. If I solve Laplace's equation in 2D and use more than 5 grid levels, the V-cycles stop converging after a few cycles (see below, convergence factor > ...
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3answers
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How to deal with curved boundary condition when using finite difference method?

I'm trying to learn about numerically solving PDE by myself. I've been beginning with finite difference method(FDM) for some time because I heard that FDM is the fundament of numerous numerical ...
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0answers
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Finite Difference for Hamilton Jacobi Belman

I have hjb equation where $V=V(x,t)$ and $u=u(x,t)$ $V_t + \sup(u) [A(x,u)V_x + B(x,u)V_{xx}]=0$ for $x$ in $[0,1]$ and $t$ in $[0,1]$ I have been able to successfuly resolve it numerically having ...
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1answer
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How to implement boundary condition in this case?

Let's start off by NUMERICALLY solving a 1-D steady-state heat transport problem using IMPLICIT FDM. $DT_{xx}=0; ~T(x=0)=T_{BL}; ~T(x=l)=T_{BR}$ where $D$ is diffusion; $T$ is temperature; subscript ...
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2answers
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Solid mechanics with finite differences: How to handle “corner nodes”?

I have a question concerning coding boundary conditions for solid mechanics (linear elasticity). In the special case I have to use finite differences (3D). I am very new to this topic, so perhaps some ...
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1answer
201 views

boundary oscillations with Robin boundary conditions

When solving Poisson's equation on the unit square $\Omega$ with homogeneous Dirichlet boundary conditions for $x=0$ and Robin-type conditions at the rest of the boundary, $$ \begin{cases} -\Delta u = ...
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3answers
4k views

Applying Dirichlet boundary conditions to the Poisson equation with finite volume method

I would like to know how Dirichlet conditions are normally applied when using the finite volume method on a cell-centered non-uniform grid, My current implementation simply imposes the boundary ...
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1answer
1k views

boundary conditions for electrostatic problem

I'm solving an electrostatic problem governed by Laplace equation $$-\nabla \cdot (\rho^{-1} \nabla u) = 0$$ in the following domain: a brick ($\Omega_1$) with a cylindrical inclusion ($\Omega_2$), ...
5
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1answer
203 views

What Linear Equation Solver should be used for a problem with many dirichlet conditions?

I am solving a laplace equation on a finite-element mesh (tetrahedral, triagonal) and have many say 99% dirichlet conditions compared to the number of unknowns. Is there an efficient way to solve this ...
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2answers
2k views

Finite-volume method: can Dirichlet boundary conditions be applied to the integral form?

I would like to apply Dirichlet conditions to the advection-diffusion equation using the finite-volume method. This answer, "How should boundary conditions be applied when using finite-volume method?" ...
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1answer
3k views

Finite Difference Method Neumann Boundary Condition with Variable Coefficients

Disclaimer In the process of typing up this question, I determine its solution. Since I went through the trouble of typing up the question in its entirety, I will post its answer as well. It may ...
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1answer
14k views

How should boundary conditions be applied when using finite-volume method?

Following from my previous question I am trying to apply boundary conditions to this non-uniform finite volume mesh, I would like to apply a Robin type boundary condition to the l.h.s. of the domain (...
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1answer
873 views

Are we free to choose the position of ghost cells on a non-uniform finite-volume mesh?

Following Hundsdorfer approach the finite volume discretisation of the advection-diffusion equation (conservative form) on non-uniform cell centered grid can be written as, $$ w_j^{\prime} = \frac{w_{...
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1answer
529 views

Implementation of a contraction force in Fenics

Is there any way to implement an element wise contraction force (i.e., a force which causes the FEs themselves to contract onto themselves)? For example this would happen when something dehydrates. ...
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0answers
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Computation of potential flow using dirichlet conditions [duplicate]

I know I already posted this question and I thank you for your answers, but unfortunately I didn't find what I was looking for among them. Anyway now I'll rewrite the question more clearly, because ...
7
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1answer
390 views

Potential flow around a non-symmetric obstacle using stream functions

I've seen that there is a way to use the finite differences method, on a cartesian orthogonal grid, to perform calculations on potential flow about an obstacle without using the Neumann conditions, ...
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2answers
2k views

FEniCS: boundary conditions for electrostatic problems with dielectrics

I carefully read all circa 70 pages of FEniCS tutorial and I still do not understand how to solve electrostatic problems when I have materials with different dielectric constant. The self contained ...
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2answers
5k views

FEniCS: how to specify boundary conditions on a circle inside 2D mesh

I would like to numerically find a mutual capacitance of two stripes of metal on the opposites sides of a cylinder. The problem is obviously a 2D Laplace equation. I would like to find the potential ...
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1answer
297 views

CFD (Fluent) define a inlet for a tidal basin

I'm still pretty new in the CFD modelling world. Can anyone advise me how to define a inlet for a tidal basin in Fluent? The water level and the velocity at the inlet vary in time due to the tide ...
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1answer
2k views

Trouble implementing Neumann boundary conditions because the ghost points cannot be eliminated

Neumann boundary conditions are implemented by introducing ghost points outside the domain and then using the boundary conditions to eliminate the ghost points. For example, see this question. I ...
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2answers
4k views

FEniCS: separate boundary conditions in normal and tangential direction of mesh boundary

Given a vector-valued PDE, I'd like to enforce the boundary conditions $$ \vec{n}\cdot u = g\\ \vec{n}\cdot \nabla (\vec{t}\cdot u) = 0 $$ on the solution $\vec{u}$. If the boundary happens to align ...
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1answer
2k views

Example of a PDE model with nonlinear Dirichlet boundary conditions

Is there any application for PDEs with nonlinear Dirichlet boundary conditions? That is, I am looking for an example of a partial differential equation for a state $u$ posed on a domain $\Omega$ with $...
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0answers
965 views

Newton Iteration method convergence

I wrote a Python code which solves a second degree nonlinear differential equation using the Newton iteration method. The code converges to a 2-cycle within 50 or so iterations. The cycle only ...
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3answers
4k views

Role of boundary conditions (e.g. periodic) in Poisson equation

Given 3D Poisson equation $$ \nabla^2 \phi(x, y, z) = f(x, y, z) $$ and the right hand side and the domain, am I free to impose any boundary conditions (BC) on the function $\phi$, or do they have to ...
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3answers
2k views

No flux boundaries for mixed hyperbolic parabolic PDE

I read this post, "Conservation of a physical quantity when using Neumann boundary conditions applied to the advection-diffusion equation" and although it is the same type of equation it does not fit ...
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3answers
3k views

Open boundary conditions with the advection-diffusion equation

Following on from my previous equation I'm would like to apply open boundary condition to the advection-diffusion equation (with reaction term), $$ \frac{\partial \phi}{\partial t} = \frac{\partial}{\...
28
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1answer
6k views

Conservation of a physical quantity when using Neumann boundary conditions applied to the advection-diffusion equation

I don't understand the different behaviour of the advection-diffusion equation when I apply different boundary conditions. My motivation is the simulation of a real physical quantity (particle density)...
3
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1answer
474 views

Marker and Cell Method (MAC) - STOKES FLOW - boundaries?

please can you help me with my problem with Stokes flow written using Marker and cell method (MAC)? I need only to solve the eq. of continuity + momentum eq. for a given condition (steady state). I ...
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1answer
302 views

mathematical statement of “open” boundary condition

For your information, the original equation comes from here. Note: You DON'T have to read the paper. I will make the question as self-contained as possible. The central equation to solve is equation (...
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2answers
19k views

Writing the Poisson equation finite-difference matrix with Neumann boundary conditions

I am interested in solving the Poisson equation using the finite-difference approach. I would like to better understand how to write the matrix equation with Neumann boundary conditions. Would someone ...
5
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3answers
262 views

which numerical method for ode with mixed BCs

I've got a second order nonlinear ODE (nothing fancy), but the BC are a little odd to me: $y'(0) = 0$ $y \rightarrow y_a$ as $x \rightarrow \infty$ What's a good numerical method for solving this? ...
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2answers
6k views

How to properly apply non-homogeneous Dirichlet boundary conditions with FEM?

In general, Dirichlet boundary conditions won't be satisfied exactly for FEM for non-homogeneous boundary conditions. The FEM codes I've seen set the degrees of freedom to interpolate the Dirichlet ...
6
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1answer
744 views

how to visualize lattice with periodic, helical, etc. boundary conditions?

I am trying to write a special hexagonal lattice generator, with several kinds of boundary conditions, such as helical BC, periodic BC, and I find it hard to verify whether it works correctly. I tried ...
4
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1answer
246 views

How do I solve an ODE Two-Point Boundary Value Problem?

I have a feeling my question is a very basic one, but I am not at all well versed in computational sciences. My equations are of the form: $$ y \in \mathbb{R}^3 \\ \dot{y}(t) = f(y(t)) \\ y_1(0) = a ...
6
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1answer
224 views

Neumann BCs in cylindrical geometry (FEM)

I was wondering where I could get a detailed account (either in print or online) on applying a Neumann/mixed Boundary condition along the $r=0$ axis in an axially symmetric geometry. Though this is a ...
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4answers
2k views

Heat equation (steady state) with boundary conditions at domain edges and inside the domain

I'm trying to solve the steady state of a heat equation problem in 2D $\Delta u = 0$ (3D also), with the method of solving the huge system of equations that arises from the discretization of the ...
5
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1answer
2k views

Mixed boundary conditions Finite Element Method

I have the following problem in Finite Element Method $$ -(\alpha u')' + \beta u' + \gamma u = f$$ with $ \Omega = (0, 1)$, $ u(0) = 0 $ and $ u'(1) = 3 $ to be able to write the weak formulation ...
14
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4answers
7k views

Boundary conditions for the advection equation discretized by a finite difference method

I am trying to find some resources to help explain how to choose boundary conditions when using finite difference methods to solve PDEs. The books and notes which I currently have access to all say ...
11
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4answers
7k views

solving coupled ODEs with initial-value and final-value constraints

The essence of my question is the following: I have a system of two ODEs. One has an initial-value constraint and the other has a final-value constraint. This can be thought of as a single system with ...
3
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2answers
429 views

What numerical methods are recommendable for simulating two phase immiscible fluid flow through a pipe with high capillary pressure?

I'm simulating two phase immiscible drainage (air displacing water) in a rectangular domain of size .6mm x 2.4mm (2 dimensions) using Ansys FLUENT software. I am using an implicit Volume of Fluid ...
24
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4answers
3k views

How to incorporate the boundary conditions with the Galerkin method?

I've been reading some resources on the web about Galerkin methods to solve PDEs, but I'm not clear about something. The following is my own account of what I have understood. Consider the following ...
8
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1answer
270 views

Adaptive mesh refinement with perfectly matched layers?

We have an adaptive mesh refinement (AMR) code for solving the elastic wave equation with frictional fault interfaces (based on Chombo for those that are interested). One of the things that we have ...

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