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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339 views

Dirichlet boundary conditions in the 1D Heat Equation

Please consider the assignment I have uploaded on the picture. I am confused about the functions $g_L(t)$, $g_R(t)$ and $\eta(x)$. What are they and how do I find them... My question: Is it possbile ...
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2answers
1k views

Boundary conditions in conjugate gradient method for poisson's equation

I want to use the conjugate gradient method to solve poisson's equation in an electrostatic setup: \begin{align} \rho=-\nabla^2\phi \end{align} I am however a little confused when it comes to the ...
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1answer
359 views

Satisfying Periodic Boundary Conditions while plotting spherical particles inside a cube

I am trying to plot spherical particles in a cube of fixed dimension in matlab. I face a problem here where the center of the sphere is too close to the edge of the cube in this case the rest of the ...
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1answer
294 views

Relation between Stress/Strain and normal derivative of displacement

I calculated the stress $\sigma$ and strain $\varepsilon$ for a solid plate with dirichlet boundary conditions $u = g$, where $u$ is the displacement. With these I want to calculate $\nabla_n u = t$ ...
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1answer
91 views

Harmonic oscillators with periodic boundary conditions

I am trying to simulate multiple harmonic oscillators in periodic boundary conditions (subsequently visualizing the process in VMD). I have successfully simulated multiple HOs by using the Leapfrog ...
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1answer
584 views

Well-posedness of linear elasticity boundary conditions

I have several questions regarding suitable boundary conditions for linear elasticity. I have read that in order for modeling linear elasticity to be well-posed, the entire boundary cannot be ...
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1answer
146 views

Parallel calculation in finite elements

I am trying to solve a 1 Dimensional eigenvalue of poisson problem: $$\nabla \phi ^2 +\nabla \phi = k\phi$$ with the boundary condition: $\phi (0)=0 , \nabla \phi(1) = 0 $. I could solve this ...
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1answer
275 views

Raviart Thomas Mixed Finite Element with Mixed boundary conditions reference request

This is perhaps a more focused version of this question. Using standard notation, I have a code that works and solves the following PDE using a Raviart Thomas mixed method. $$\begin{align} 0 &= ...
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1answer
842 views

Flux boundary condition in solute transport

I have a pretty naive question, though important to me. Usually when solving the following PDE in solute transport: $\frac{{\partial C}}{\partial t } = \nabla. (D\nabla C -vC )=0,$ one can be asked to ...
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1answer
86 views

Perfectly matched layer simulation with two vibrating sources

I'm doing PML simulation, and while I was looking for correct geometry for my layers of propagation, I got a question. My device has two vibration sources, and this vibration will propagate through ...
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2answers
112 views

Definition of inflow boundary in CFD

If $w$ is the vector of conservative variables, $f=f(w)$ the flux function, I think have read somewhere (I can not find it anymore) that the inflow boundary $\Sigma$_ is characterized by: $\Sigma_{-}...
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1answer
102 views

Finite difference scheme for Webster equation

Webster equation is a popular generalization of 1D wave equation used for ducts of variable cross-section $S \equiv S(x)$. Assuming harmonicity in time, the spatial equation for propagation of ...
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2answers
177 views

BCs in a coupled problem

Consider a thermo-mechanical coupled problem, where coupling exists from both the sides, mechanical loading producing thermal effects and vice versa. In such a case, is it necessary to always ...
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2answers
270 views

Dirichlet BCs - alternative implementation methods

I am having problems with solving a hyperbolic wave problem with Dirichlet BCs. I have tried reducing the time step sizes, which does not affect the results, and notices increasing the number of nodes ...
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1answer
230 views

Please explain the meaning of these Boundary conditions [closed]

I am trying to learn Gmsh and Fenics and was looking at an example which shows the application of Boundary conditions on a simple Poisson problem. Here is the link: http://fenicsproject.org/...
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1answer
86 views

methods for a peculiar BVP system

Consider the following system defined on the open interval (-1, 1): $y_1' = c y_3 \\ y_2' = c y_4 \\ y_3' = -f(y_1, y_2)y_2 \\ y_4' = f(y_1, y_2)y_1 $ given $ y_3(-1) = 0 = y_3(1) \\ ...
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1answer
287 views

What kind of test cases are convenient to use for testing the code for Euler equations of gas dynamics in polar coordinates?

Consider Euler equation of gas dynamics in polar coordinates as $$ \left( \begin{array}{ccc} \rho \\ \rho u_{r} \\ \rho u_{\theta} \\ E \end{array} \right)_{t} + \frac{1}{r} \left( \begin{array}{ccc}...
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1answer
63 views

Where could error terms that blow up in SWE come from?

I have been working on a solver for shallow water equations with reflective boundary conditions. I have found that it diverges very fast. As a workaround I noticed yesterday that if I smooth the ...
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1answer
111 views

traction boundary conditions in elasticity

I have a question about implementing traction boundary conditions in 2D and 3D linear elasticity. Consider the picture above. I want to apply traction boundary conditions on the boundary in red. My ...
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1answer
74 views

Simulating pressure waves at an impedance boundary

I am trying to simulate pressure waves crossing a boundary from one medium to another (e.g., water to air) in Matlab. The code that I have got so far, which is largely taken from Wikipedia on Partial ...
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1answer
83 views

FEM diffusion: inaccurate results small time steps

I wrote some FEM code and found some strange results when using a very small time step. So, I decided to analyze the discrete equations. Consider the following linear diffusion problem in 1 dimension:...
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1answer
93 views

Dealing with boundary conditions using Fourier spectral methods

I am currently working on a project where I need to use Fourier spectral methods to solve the KS equation. I found this code which is using the Fourier spectral methods to solve the classic 1D heat ...
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1answer
462 views

Implementing Robin Boundary condition (finite difference)

I'm interested in applying Robin boundary condition to a convection-diffusion problem in 1D. In the following system, $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - v\frac{\...
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1answer
288 views

Are mixed boundary conditions possible in structural mechanics?

For structural mechanics, such as linear elasticity, I am aware of BCs such as a prescribed displacement (Dirichlet) or a prescribed traction (Neumann). Is it possible that a boundary can have a ...
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1answer
798 views

Proper boundary conditions for potential flow around cylinder

I am computing the stationary, incompressible, inviscid and irrotational flow around a circular cylinder using a discretization in general coordinates. I derived a PDE and proper boundary conditions ...
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4answers
320 views

Mass transport in porous media

My goal is to numerically solve the convection-diffusion equation of the form: $$\frac{\partial C}{\partial t} = \nabla (D \nabla C) - \nabla (v C)$$ $C$ is concentration and $D$ is the diffusivity ...
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1answer
2k views

OpenFoam Mapped Boundary condition [closed]

I am using OpenFoam to do an LES simulation. I am using mapped boundary condition on the inlet plane. I know it maps from a source plane to the target plane(here the inlet plane). In "Pitzdaily-mapped"...
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1answer
296 views

FD implementation of Absorbing Boundary condition for acoustic wave

I am simulating acoustic wave equation in which absorbing boundary condition has to be applied. It is applied in two ways Ist method is as mentioned in this paper. Boundary condition at bottom is (...
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1answer
253 views

imposing “measured data” to Dirichlet boundary conditions in fenics

I'm relatively new to fenics and I just looked through all questions related to Dirichlet boundary conditions. I don't seem to find a well-described question or answer about what I'm about to ask. I'...
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1answer
400 views

Outflow boundary condition

I know that in outflow boundary we assume a zero normal gradient condition and use upwind scheme for approximation. However, I saw this sentence in a book which I do not understand; "Convective fluxes ...
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1answer
789 views

oscillating flow inlet BC

I want to do LES of an oscillating flow i.e. a sinusoidal flow without a mean component in a channel. In order to have a fully developed flow at lower mesh count without using a long channel I want to ...
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1answer
1k views

Instability of pdepe in Matlab… boundary conditions?

here is a Matlab beginner banging his head on the wall... I am trying to solve a system of partial differential equations in Matlab, with both derivatives in time and space domains. I am using the ...
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1answer
3k views

Implement Robin Boundary Condition

This is a follow up to this question. I have a nonlinear BVP on $x=0$ to $L$: $$ (T^2)\frac{\partial^2 T}{\partial x^2} + T \left( \frac{\partial T}{\partial x}\right)^2 + Q = 0 $$ to which I apply a ...
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1answer
794 views

Euler's equations 1d for pipe, Inlet boundary conditions

$\def\rmin{{\mathrm{in}}}$ $\def\l{\left}\def\r{\right}$ $\def\tagl#1{\tag{#1}\label{#1}}$ I am using the one-dimensional finite volume method to calculate the air flow in some tube. For subsonic ...
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0answers
52 views

Constrained optimization for non-linear equations in octaveGNU

I have installed Optim1.6.1 package. I would like to solve a system of equations in non linear finite element analysis using constraints as u=1 at certain nodes. u=0 at certain nodes. Typically I find ...
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0answers
119 views

Weak form of Elliptic problem with mixed Dirichlet & Neumann conditions

Let $\Omega \subset \mathbb{R}$ in a bounded polygon domain and $f:\Omega \to \mathbb{R}$ known function.We split the boundary into two parts $\partial \Omega_{1}$ and $\partial \Omega_{2}$ such that $...
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0answers
47 views

Why is this Advection-convection model with insulating boundary losing mass?

I am trying to model a 1-d advection-convection numerically, using an upwind scheme. I'm using the following equation to calculate the value of internal cells: $$C_x^{t+1} = C_x^{t} + D\frac{\Delta t}{...
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0answers
54 views

Solving Laplace equation with constraint on boundary

I have found the following PDE problem in a paper: Essentially, we have a rectangular domain where there is a unknown interface $z=\xi(x)$ (liquid-air interface) separating the domain into two medium ...
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0answers
46 views

Integrating a wavelike equation with absorbing boundary conditions

I am trying to numerically solve the following equation: $\frac{\partial^{2} \phi}{\partial t^{2}}-\frac{\partial^{2} \phi}{\partial x^{2}}+V(x) \phi(x, t)=0$ On some domain, with: $\phi(x, 0) = I(x)$ ...
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38 views

Discretization formula for a system of two differential equations. “Solution to one of these is the initial condition of the other”. In which sense?

Consider the following stochastic differential equation \begin{equation} dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1} \end{equation} where $A$, $B$ and $C$ are parameters ...
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0answers
42 views

Euler Explict Method for solving the PDE for option prices under the Schwartz mean reverting model. Numerical Finance

I have to solve the following PDE for a Call option : $\partial_tV + \{ \alpha - (\mu - \lambda/ \alpha -log(S))\}S\partial_SV + 1/2 \sigma^{2}S^{2}\partial_{S}^{2}V - rV = 0$ Where V(S,t) is the ...
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40 views

Effect of reducing flux consistency order at boundary on convergence order

Consider the 1D nonstationary convection-diffusion PDE $$ \begin{alignat}{2} \partial_t u &= -a \partial_x u + D \partial_{xx}u, &\qquad x \in (0,1), t \in (0,T), \\ f(t) &= \left.\left( a ...
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0answers
92 views

Linearize non-linear PDE with BCs to hyperbolic problem: How does linearization affect BCs?

I am working with the Shallow Water equations that is a system of non-linear PDEs that simulate water waves propagation on some domain, in my cases the $x$-axis. I have here a GIF showing the results (...
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0answers
141 views

Incorporating radiation boundary condition at the edge in finite difference

I am trying to solve the 2-d heat equation on a rectangle using finite difference method. I am confused as to how to incorporate non linear radiation boundary condition at the edge. $-k\frac{\partial ...
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37 views

How to avoid density getting “deleted” in two way rigid body coupling with LBM CFD?

I've been reading this paper recently, which talks about using Lattice Boltzmann methods and two way coupling. Specifically, it outlines fluid solid coupling, and solid fluid coupling, and how simply ...
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0answers
74 views

Solving a system of PDEs with no-flux boundary conditions (finite difference discretization)

I am interested in solving a system of linear PDEs with the finite difference method and I'm having trouble to solve the no-flux boundary condition correctly. \begin{align} \frac{\partial n}{\partial ...
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0answers
70 views

Solving diffusion equation using finite difference method

I am solving an 1-dimensional diffusion equation with Neumann boundary condition at outlet and constant concentration, C, at the inlet. In the end, I want to observe how the concentration diffuses ...
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0answers
108 views

FDM discretization of equation on the boundary

In order to simulate the following equation using FDM $$u_t(t,x)-u_{xx}(t,x)=0, \quad (t,x) \in (0,1)\times (0,1)$$ $$(u_t(t,x)-u_{x}(t,x))\rvert_{x=0}=0, \quad t \in (0,1)$$ $$(u_t(t,x)+u_{x}(t,x))\...
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0answers
38 views

Boundary conditions for a Non-linear Schrödinger equation using an extended crank nicolson scheme

I try to solve numerically the following PDE for $E(r, z)$ with a cylindrical symmetrie (i. e. $E(r, z) = E(-r, z)$). $\frac{\partial E}{\partial z} = \frac{i}{2k} \Delta E + \mathcal{N}(E)$ Where $...
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114 views

PDE discretization on triangular domain

Given the 2D Poisson equation $$\Delta u = f\\ u(x,0) = g_1(x), 0<x<1\\u(0,y) = g_2(y), 0<y<1\\ \partial_n u (x, 1-x) =0, 0<x<1$$ defined on the domain $\Omega := \{(x,y) \in \...

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