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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2
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1answer
63 views

1-D boundary value problem: How implement mixed boundary conditions using a FD method?

I have been given a convection-diffusion ODE modeling the steady state temperature of a pipe (through which flows a fluid) as $$-\frac{d}{dz}\left(\kappa \frac{dT}{dz} \right)+v\rho C\frac{dT}{dz}=Q(...
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1answer
64 views

Implementing structured grid boundary conditions using NumPy arrays?

I am making a toy code in Python to solve the advection equation $$u_t + cu_x = 0$$ with, for example, periodic boundary conditions. Background information The numerical grid is specified like this: ...
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0answers
63 views

How to implement the following Finite Element method for Burgers' equation?

I am trying to replicate this result. It involves using the Galerkin finite element approach onto the viscous Burgers' equation. However, my implementation (in R) seems to be giving me wrong results....
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1answer
83 views

Well-posedness of Navier-Stokes equation

Before running a simulation for turbulence (e.g Rayleigh-Benard Convection), how do we check for well-posedness of Navier-Stokes with other equations for a given boundary condition?? Can someone ...
2
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0answers
80 views

Implementation of boundary conditions for 1D Euler equations

I'm trying to solve 1D Euler equations with gravity in spherical coordinates using a finite-difference TVD MacCormack method on a non-uniform grid of $N$ components, following the method provided in ...
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0answers
60 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
46 views

A non linear ode with boundary conditions at infinity

I want to solve the non-linear ODE $$\frac{d^2}{dx^2}y=a(y+y^3)$$ With the boundary conditions that $$\lim_{x\to \pm \infty} y(x) =0$$ I am not aware of any analytical method for solving this kind ...
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1answer
82 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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0answers
36 views

Neumann boundary conditions on arbitrary surface for finite difference diffusion

I am facing the following problem, formulated in practical terms: I have a region $\Omega$ in two or three dimensions, represented as a binary mask, and an initial density $u_0$ within that region ...
0
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1answer
68 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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0answers
31 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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0answers
36 views

Boundary conditions for solving the time-independent SE for the hydrogen atom

I am trying to solve the schrodinger equation for the hydrogen atom numerically, using finite elements, with matlab's solvepdeeig(). I have a hard time getting the solution to be right, and it seems ...
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1answer
73 views

Imposition of Dirichlet BC for Fourier pseudospectral in this paper

I was trying to implement the algorithm from the paper "Adapting a Fourier pseudospectral method to Dirichlet boundary conditions for Rayleigh–Benard convection". I am having a hard time to ...
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2answers
134 views

Non-linear Boundary Value Problem. How to compute the Jacobian?

Consider a Boundary Value Problem: $$ \delta u''+u(u'-1) =0 \Leftrightarrow u''=\frac{-u(u'-1)}{\delta}=:f(t,u',u), \\ u(0)=a, u(1)=b $$ $\delta,a,b$ are known parameters. I want to implement Newton'...
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0answers
46 views

Existence and uniquness of solution of FVM for Poisson equation

I'm discretizing the following Poisson equation using FVM where the domain $\Omega$ of the solution is a regular hexagon of side $1$ centered about the origin. $$\Delta u =k,\text{ $k$ constant}\\ \...
2
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1answer
51 views

Linear elasticity modeling load using traction vs. mixed BC

In classical linear elasticity, when modeling a force/load boundary condition, it appears that we could either: Use a pure Neumann boundary condition, where the 3 traction components are specified. ...
1
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1answer
45 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 ...
0
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1answer
61 views

Structural boundary conditions - rotational/translational DoFs and displacement/tractions BCs

I am a little bit confused over the concept of translational and rotational degrees of freedom (DoFs) in structures, and their relation to displacement/traction BCs. Do displacement boundary ...
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0answers
30 views

Boundary Conditions involving exponential functions of nodal unknowns

I am fairly new to Computational Engineering and I have mainly been exposed to using the Finite Difference Method to produce Linear Systems and solve them using Iterative Methods. I am trying to ...
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0answers
23 views

Non linear Parametric BVP with inequalities

Consider a non linear ode in dimension $10$: $\dot x = f(t,x,\lambda)$ where $\lambda$ is a vector of $p$ parameters. Consider a boundary value problem of the form : $\dot x(t) = f(t,x(t),\lambda)$ ...
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1answer
80 views

Combining multiple coupled 1st order equations in python

I'm having serious troubles with solving translating 3 coupled differential equations into python. The 3 DE's stem from a 4th order DE used to calculate the bending moment of an underwater pipeline ...
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0answers
78 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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1answer
150 views

Discretization Neumann boundary condition

I'm currently working with the following Poisson equation with mixed boundary conditions, including a Neumann boundary condition. $$\Delta u = f\\ u(x,0) = g_1(x), 0<x<1\\u(0,y) = g_2(y), 0<...
0
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1answer
171 views

Heat diffusion - Is this the correct approach to include Newmann boundary conditions?

Thank you for looking at this problem. Is this the correct approach to include neumann boundary conditions? With this solution temperature is not correct, and there´s no diffusion. The model seems ...
2
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0answers
58 views

Unusual boundary conditions on Matlab

I'm trying to solve the following PDE by Matlab, $$ u_t-\Delta u = 0, \quad \text{in}\quad \Omega\times (0,T) \tag{1} $$ $$ u_t-\Delta_\Gamma u + \partial_\nu u=0,\quad \text{on}\quad \Gamma\times(0,...
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0answers
30 views

Hydrogen-like wavefunction as starting guess for atomic solver?

I've been looking into radial solvers for quantum wave equations (Schroedinger and Dirac). In both cases, the suggestion seems to be to go with the "shooting method", with integration schemes of ...
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0answers
65 views

Implementing boundary condition

I'm studying the transport of species A in the blood vessels, $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$ At x=0, I want to use the ...
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1answer
400 views

Solving advection equation - periodic conditions - using roll python function [closed]

The original post was on stackoverflow : I transfert it here. I have to solve numerically the advection equation with periodic boundaries conditions : u(t,0) = u(t,L) with L the length of system to ...
2
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0answers
82 views

Are there known accuracy issues between 2D axisymmetric and 3D solutions?

In my full 3D solutions I am solving for the potential throughout a $100\times 200\times 200$ grid. Inside is a ring electrode set to -5V via a Dirichlet boundary condition, and surrounded on all ...
0
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1answer
266 views

Impose Neumann Boundary Condition in advection-diffusion equation 1D

when solving the advection equation in 1D that is: $$ \frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0 $$ with $ u'(t,0) = 0$ and $u(t,L) = 0$ , $u(0,x) = u_{0} $ one numerical ...
0
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1answer
880 views

Applying neumann boundary conditions to diffusion equation solution in python [duplicate]

For the diffusion equation $$ \frac{\partial u(x,t)}{\partial t} = D \frac{\partial ^2 u(x,t)}{\partial x^2} + Cu(x,t) $$ with the boundary conditions $u(-\frac{L}{2},t)=u(\frac{L}{2},t)=0$ I've ...
1
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1answer
111 views

Shooting method implementation

I have a trouble of defining a Jacobian matrix for my problem. Basically, I have 4 differential equations to be solved. $$ \begin{aligned} \dot x_1(t)&=x_2(t)\\ \dot x_2(t)&=p_2(t)−\sqrt 2 ...
2
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1answer
72 views

PML boundary conditions

I set up two one-way wave equations for constant velocity $c$ in one-dimension. When I implement them I get a highly unstable (divergent) solution. I wonder if someone could give me a suggestion about ...
2
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0answers
66 views

Neumann boundary conditions in the Maccormack scheme

I am trying to solve the viscous Burger equation $$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \mu \frac{\partial^2 u}{\partial x^2} $$ with Neumann boundary conditions. I am ...
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0answers
131 views

Inflow and outflow boundary conditions for advection-diffusion equation

I'm trying to solve this advection-diffusion equation (ADE): $$\frac{\partial \phi}{\partial t} + \nabla \cdot (-D \nabla \phi + \mathbf{u} \phi) = 0$$ In fact, this ADE framework is coupled to a ...
2
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0answers
44 views

Finite difference Neumann boundary conditions: uneven weighting of edge nodes?

Originally asked this on math.stackexchange, but I figure it's also appropriate here. I'm reading through some finite difference code for a diffusion equation and came across something odd for the ...
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0answers
47 views

What is the finite-difference representation of the Laplacian operator with periodic boundary conditions? [duplicate]

I am using a central-difference scheme to solve the eigenvalue problem $$\frac{d^2}{dx^2}u = \lambda u$$ on a unit interval with periodic boundary conditions. My understanding is the eigenvectors $...
1
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1answer
641 views

Interatomic distance-periodic boundary conditions-non cubic unit cell

I am trying to find interatomic distance considering periodic boundary conditions for hexagon cubic cells (graphite). I tried to follow the answers to these two questions here but am unable to get the ...
0
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1answer
125 views

Boundary conditions in a four point bend test

I am looking into the four-point bend test, such as one in this YouTube video. Sample screenshot from the video illustrating the problem: I am a little confused as to how the loads are prescribed ...
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0answers
51 views

Decrease in slope during convergence analysis

I am using the method of manufactured solutions to perform the order of accuracy testing. I am using a cube for the testing. The cube is size 1m on all sides. I used 5 refinements: $dx = dy = dz = ...
1
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1answer
276 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 ...
1
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0answers
63 views

Method of manufactured solutions - choice of type of boundary conditions

I am attempting to use the method of manufactured solutions (MMS) for code verification for linear elasticity. However, this is more of a general question regarding the general use of MMS. In MMS, ...
2
votes
1answer
92 views

FEM-Laplace with Dirichlet in only a few points: Nonsingular operator?

Let's consider the FEM discretization of the Laplace operator without boundary conditions, i.e., $$ a(u,v) = \int_\Omega \nabla u \cdot \nabla v - \int_\Gamma (n\cdot \nabla u) v. $$ For one-...
1
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1answer
170 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 ...
0
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1answer
177 views

Proving solution existence and uniqueness of the Helmholtz equation with Robin boundary conditions with complex coefficients

I am trying to solve the Helmholtz equation with Robin boundary conditions with complex coefficients and the weak formulation $$ \iint_\limits\Omega\nabla p_0(x,y)\nabla\left(\overline{v(x,y)}\right)...
2
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1answer
457 views

Traction -> stress; stress->displacement gradient

If given a displacement gradient tensor, we can easily obtain the stress tensor (using Hooke's law and the strain-displacement relationship), as well as the traction vector. If given a traction, and ...
2
votes
1answer
103 views

Closed boundary conditions in finite difference method for diffusive-advective equation

I am implementing a finite difference method in solving the diffusive-advective equation: $$ u_t + v \cdot u_x = D\cdot u_{xx} $$ (v, D are constants). Planning to use the operator splitting method (...
1
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0answers
149 views

Neumann boundary condition FD implementation for instationnary diffusion equation

I am trying to solve this diffusion equation : $\dfrac{\partial D\dfrac{\partial f}{\partial x}}{\partial x}+S = \dfrac{\partial f}{\partial t}$ ($D$ is not constant and varies according to $x$) with ...
3
votes
2answers
586 views

Implementing no-flux boundary condition reaction-diffusion PDE

I'm having trouble figuring out how to implement boundary conditions for this problem: \begin{align} \frac{\partial n}{\partial t} &= D_n\nabla^2n - \nabla\cdot\left(\frac{\chi}{1+\alpha c}n\nabla ...
2
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0answers
61 views

Finite differenced eigenvalue prob. of inhomogeneous boundary conditions?

I am basically asking about eigenvalue problems of differential equations using some finite difference method (FDM). Usually the system is subject to some boundary conditions (BC), e.g., Dirichlet or ...