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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926 views

Solving Stokes flow with walls using Oseen tensor

Introduction I've developed a code to solve for generalised, incompressible 2D Stokes flow $\eta \nabla^2 \mathbf{v} - \nabla p + \mathbf{S} = 0$ $\nabla . \mathbf{v} = 0$ where $\mathbf{S}$ can ...
3
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1answer
620 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). ...
3
votes
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 = ...
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 ...
3
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1answer
376 views

How to reformulate a 1/x^2 singular term to 1/x so that bvp4c can solve it?

I have a ''cosmetic'' problem with a singular term in my Matlab script. I am trying to solve the following system of differential equations: $$ \begin{aligned} y_1' &= y_2,\\ y_2' &= \frac{...
3
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2answers
1k views

Dirichlet boundary condition for sparse matrix - Solving Ax=b only for free nodes?

I am solving Biot equation with sparse matrix in MATLAB. I have no problem with global sparse matrices assembly, but when I assign Dirichlet boundary condition, it is so slow. From this topic, How to ...
3
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2answers
291 views

Understanding Neumann BC

I understand what Neumann BC means physically and how to imply them. However, I am not able to perfectly understand it the way it is represented mathematically as $\partial u / \partial n$ where $n$ ...
3
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2answers
854 views

Get symmetric Finite Difference matrix in non Laplacian settings

I would like to solve a system of differential equations $u+\nabla(\nabla\cdot u)=f$ or in more detail $a+\partial_t^2a+\partial_t\partial_xb+\partial_t\partial_yc=f$ $b+\partial_x^2b+\partial_x\...
3
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1answer
306 views

steady state solution from parabolic problem vs solution of elliptic problem

My question is related, but not a duplicate of Asymptotic convergence of the solution to a parabolic pde to the solution of an elliptic pde Suppose I solve the parabolic PDE: $u_t = \Delta u + f(x,t)...
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 ...
3
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1answer
347 views

Boundary conditions in conforming Galerkin method for biharmonic equation

I am trying to solve simple scalar biharmonic equation using bubnov-galerkin finite element method. I am using $H^2$ conforming basis functions. I was wondering that if anyone can give me some ...
3
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1answer
151 views

Imposing symmetry plane boundary condtition

I want to impose symmetry plane boundary condition for a solid mechanics problem. I googled around and found out that in many places people say to "forbid displacemnts out of symmentry plane and ...
3
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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} + ...
3
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2answers
1k 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 ...
3
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1answer
457 views

Periodic boundary condition in solid

I want to solve a small deformation solid structure problem applying periodic boundary conditions in FEM. The geometry is a square and the equations are: $$ \text{div} \, \sigma = 0 \\ \sigma = f(\...
3
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1answer
283 views

Pressure-Pressure BC

My FDM code simulates Backward Facing Step flow when I use conventional BCs such as defining velocity profile at inlet and fully developed condition at outlet. I have validated the results and it ...
3
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2answers
433 views

2D Laplace problem with mixed boundary conditions using Conjugate Gradients

I am being asked for one of my classes to solve 2D Laplace equations with mixed boundary conditions using the Conjugate Gradient method. The equations and conditions are given as: $$ \frac{\partial^...
3
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1answer
220 views

LU-SGS and boundary conditions

I am trying to understand how boundary conditions are implemented when one uses the nonlinear LU-SGS algorithm for Euler equations. Most papers describe the Gauss-Seidel sweep over mesh cells, but do ...
3
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1answer
814 views

Smoothed particle hydrodynamics bounduary (ghost particles) properties

I am learning SPH method. At the moment I am trying to implement simulation described in this very good article. However I don't get how the ghost particles properties are computed: Position of the ...
3
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2answers
523 views

How to implement boundary conditions in heat equation with no flux and fixed value at the same time? Is it Robyn BC?

I am modeling the temperature of the groundwater using heat equation. I have Dirichlet BC at the top but at the bottom I have constant temperature equal 12 degrees C (see attached pic). It is look ...
3
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2answers
165 views

Boundary conditions for solving Poisson's Equation with Experimental Data

I want to numerically (with Matlab) solve Poisson's equation : $ \frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = f(x,y)$ On a rectangular domain using experimental data. From ...
3
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1answer
531 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. ...
3
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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 ...
3
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0answers
159 views

Non-reflecting boundary conditions for compressible Navier-Stokes equations

I have some questions about the implementation of non-reflecting OUTFLOW boundary condition for Navier Stokes equations. Following Poinsot, Lele "Boundary Conditions for Direct Simulations of ...
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0answers
505 views

Neumann-Neumann boundary intersection

I am trying to solve a Vertex Centered, Finite Difference, Poisson equation $-\nabla^{2}u=f$ with Dirichlet boundaries on the left and bottom and Neumann boundaries on the top and right using ...
3
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0answers
185 views

Multigrid for Robin boundary conditions

I have a question regarding the treatment of Robin boundary conditions in a multigrid solver. I am solving the Poisson equation in $\Omega=(0,1)^2$ with Robin boundary conditions on the boundary, $$- \...
3
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0answers
471 views

Boundary equations for constant right hand side in Poisson equation (FD)

I am setting up to solve a 2D Poisson equation $\nabla^2u = T$ by the finite difference method. I am using the multigrid method. On the coarsest mesh, where the grid size is 2x2, I want to set up ...
3
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0answers
68 views

Absorbing BC's / PML on a graph

The wave equation, $$\ddot{u} = c^2 \Delta u,$$ can be generalized to abstract graphs by using the negative graph Laplacian in place of the physical Laplacian. Is there a graph-theoretic analog of ...
3
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0answers
55 views

Discretizing boundary conditions for vortex methods

I am working on a fluid simulation using vortex methods. For this I must compute the vortex sheet on my boundaries given as: $$ \gamma(\mathbf{x}) - \frac{1}{\pi}\int_S \frac{\partial}{\partial\mathbf{...
3
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0answers
360 views

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 ...
2
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1answer
254 views

How do I simulate an open end?

When simulating a partial differential equation describing a physical phenomenon like vibrations on a string, fluid flow in a chamber or quantum wave functions, the most straight-forward way is to ...
2
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1answer
711 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
2answers
293 views

$O(h^2)$ convergence for Elliptic PDE

I am trying to solve an elliptic PDE in 2-D: $$-\nabla^{2} u = 20tanh(10x-5)(10-10tanh(10x-5)^2) = f$$ I know that the solution is $u = tanh(10x-5)$ but I am unable to get $O(h^2)$ solution with a ...
2
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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 ...
2
votes
2answers
138 views

Two-dimensional heat equation with Neumann boundary conditions: any hope to find an analytical solution?

I am looking for references showing how to analytically solve the heat equation with Neumann boundary conditions in two dimensions. So far, I have found the problem solved analytically in one ...
2
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1answer
516 views

What are acceptable boundary conditions for porous media flow?

I am attempting to simulate fluid flow through a porous foam. I would like to have no-slip boundary conditions on part of the boundary and free flow conditions on the inlet and outlet. Right now I am ...
2
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1answer
330 views

How to impose an integral conservation in solving ODEs boundary value problems (BVP)?

I have a system of coupled ODEs that I want to solve. The functions are A(x), B(x), C(x). It is a boundary values problem. I am using Matlab bvp4c. So far I am not satisfied with my solutions. For ...
2
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1answer
2k views

Solving the Poisson equation with Neumann Boundary Conditions - Finite Difference, BiCGSTAB

I am trying to solve the Poisson equation in a rectangular domain using a finite difference scheme with a rectangular mesh. I have happily generated the matrix system of equations Ax = b which is ...
2
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1answer
231 views

Neumann / natural BCs in FEA

I'm trying to work out an as-general-as-possible 2D Laplace example for Finite Element Analysis. Starting from $\Delta u = 0$ for an unknown $u(x,y)$, I multiply both sides (well, in practice only the ...
2
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1answer
774 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 ...
2
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1answer
102 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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1answer
1k views

Line integral along the edge of an isoparametrically mapped triangle

I need to integrate the following function on the line segment from $P_{1} = \begin{bmatrix} -2\\-1 \end{bmatrix}$ to $P_{2} = \begin{bmatrix} 1\\2 \end{bmatrix}$: $$\int_{P_{1}}^{P_{2}} 4x + y \ ds$$...
2
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1answer
158 views

How to determine if Kelvin-Voigt elements are dissipating stress correctly?

I am using a program that creates viscous/absorbing boundaries by implementing Kelvin-Voigt elements. The theory behind 1-D Kelvin Voigt elements is given in this Wikipedia page. In my case I am ...
2
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1answer
246 views

Imposing boundary conditions for PDE quadratic eigenvalue problem

I have a quadratic eigenvalue problem of the form: $$(A_2 s^2 + A_1 s + A_0)\hat{v} = 0$$ where $s$ is the eigenvalue. The matrices $A_i$ contain derivatives up to order six, and I have six boundary ...
2
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1answer
1k views

Numerical method for a BVP with mixed boundary conditions (MATLAB)

I've been given a second-order non-linear ODE: $$\frac{d^{2}\theta(s)}{ds^{2}} = sf_{g}\cos{\theta} + sf_{x}\cos{\phi}\sin{\theta}$$ where $f_{g}, f_{x}$ and $\phi$ are constants. The boundary ...
2
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1answer
521 views

How can I evaluate more accurate energy eigenvalues from Schrodinger equation using shooting method?

I am trying to use the "shooting method" for solving Schrodinger's equation for a reasonably arbitrary potential in 1D. But the eigenvalues so evaluated in the case of potentials that do not have hard ...
2
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1answer
88 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. ...
2
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1answer
120 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-...
2
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2answers
193 views

Find classical solution of transport equation with FDM

We know the classical solution of transport equation is determined by one initial (boundary?) condition, for example, the solution of $$\frac{\partial u(t,x)}{\partial t}+\frac{\partial u(t,x)}{\...
2
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1answer
1k views

heat equation on bounded and unbounded domain

I have been reading about the heat equation and I am confused about uniqueness in the case when the domain is bounded and when it is not. In the book I am following, it is common to write the heat ...

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