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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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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 ...
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48 views

Solid mechanics codes mostly use Dirichlet and traction BCs - why?

In a lot of the computational solid mechanics papers that I've come across, boundary conditions are typically implemented as a traction boundary condition or a Dirichlet boundary condition. But in ...
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48 views

Wrapped BVP with an unknown boundary for fluid modeling

I have a model about a fluid being extruded on a moving bed as in the 3D printing process. The model is a boundary-value problem where the right boundary is the point where the fluid attaches to the ...
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55 views

Coupling multiple structural components without introducing numerical problems

Is there a recommended way of coupling together multiple sets of nonlinear equations representing structural components that does not cause problems for the solver? I believe this may have something ...
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51 views

How is a structural symmetry boundary condition implemented?

In link It states that: "In solid mechanics, the general rule for a symmetry displacement condition is that the displacement vector component perpendicular to the plane is zero and the rotational ...
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0answers
19 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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39 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 ...
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29 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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43 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 $...
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1answer
138 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 ...
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65 views

Neumann Boundary with ADI-method (FDM); how do I implement this?

I am modelling the Heat equation in 2D in Python. I am using finite difference methods, more specifically the Alternating Direction Implicit method. The model works quite well with Dirichlet boundary ...
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26 views

CGFEM - Adding a solvent flux to a 1D adv-diff system as a source term

I have a system consisting of a narrow pipe with porous walls where the inlet conditions are flow rate and initial concentration, and the goal is to determine the change in concentration along the ...
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1answer
56 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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21 views

Enforced (prescribed) displacements at more than 1 node of FE model [duplicate]

If I have a structural finite element model (could be continuum or frame elements), I was wondering if there is a way to enforce a prescribed displacement at more than 1 node in the model in a ...
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38 views

How does atmospheric pressure correspond to a traction BC for structural mechanics?

If you have some object that's just sitting on a table, some of its surfaces will be exposed to the ambient conditions, say at standard pressure and temperature. In computational modeling, would you ...
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0answers
44 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 = ...
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1answer
66 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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40 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
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1answer
68 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-...
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1answer
70 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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44 views

Applying base excitation to a MATLAB state-space

I have a state space model that was provided to me by exporting it from an external FEA program. The model can be described as $\dot x = Ax + Bu$ $y = Cx + Du$ This model assumes forces and ...
0
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1answer
106 views

how i can prove the exists and unique of the solution of the Helmothz equation with a robin boundary condition with complex coeficient

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)...
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1answer
128 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 ...
3
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1answer
56 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 (...
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66 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 ...
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2answers
307 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 ...
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0answers
52 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 ...
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78 views

Should ghost cells/nodes be coupled?

This is more of a theoretical question regarding the concept of ghost cells. When handling Neumann boundary conditions, ghost cells (in FVM) or nodes (in FD) are typically introduced. Essentially, ...
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1answer
183 views

Applying Neumann boundaries to Crank-Nicolson solution in python

Consider the heat equation $$u_t = \kappa u_{xx}$$ with boundary conditions of $$u(x,0)=0\\ u(0,t)=100\\ u(l,t)=0$$ Numerical analysis by pyton can be done with ...
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0answers
33 views

Can I expect reasonable results when expanding in scaled Legendre polynomials?

Imagine I want to compute the eigenvalues of an operator $\hat O$ defined on $ L^2(\mathbb{R})$, however using a properly scaled N-dimensional polynomial basis of $ L^2([-a,a]) $ which fulfills ...
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0answers
50 views

Radially symmetric system of PDEs in deal.II

I am trying to solve the radially symmetric polar form of the PDE with homogeneous Neumann BC in deal.II on a unit circle: $$ u_t = \Delta u - \nabla \cdot (u \nabla h) $$ $$ h_t = \Delta h $$ I am ...
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0answers
71 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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2answers
99 views

Second derivative in coordinate invariant form

To solve stationary, incompressible, inviscid and irrotational flow around a circular cylinder, I am using general coordinates. Since the flow is symmetrical, we only consider the upper half of the ...
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2answers
116 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)}{\...
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2answers
63 views

Finite difference for 2nd order ode $y'^2+y y''+\frac{2}{x} y y' -0.1 y^2=0$ with $y'(1)=0$ and $y(1)=1$

How to solve second order non-linear ODE $$y'^2+y y''+\frac{2}{x} y y' -0.1 y^2=0$$ subject to $y'(1)=0$ and $y(1)=1$ over the interval $0 < x \le 1$. I turned the equation to a PDE $y'^2+y y''+\...
4
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1answer
146 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(\...
0
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1answer
260 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 ...
6
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3answers
185 views

Why naively chopped finite difference matrix works for different ODE boundary conditions

We know finite difference method (FDM) can replace $y''(x)$ as $\frac{1}{h^2}[y(x+h)+y(x-h)-2y(x)]$ or so. One naive way to write down the matrix of the differential operator is like the following, ...
3
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1answer
166 views

Radiation boundary condition (heat transfer)

I am looking for reference on how to implement nonlinear boundary conditions. Specifically, I am interested in implementing a radiation boundary condition for heat transfer with the FEM: $-k \frac {\...
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0answers
103 views

How to apply an integrated constrain condition in FEM?

I'm running some simulation using FEM. In my model I need to apply a constraint condition to the governing equation. My governing equation similar to the diffusion equation as below: $$\frac{\...
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1answer
556 views

Structural mechanics - traction free = Zero displacement gradient?

This is a follow up question to my question yesterday Structural mechanics traction boundary condition question Does a traction free boundary condition also mean that the displacement gradient is 0 ...
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0answers
125 views

Structural mechanics traction boundary condition question

In structural mechanics, are the boundary conditions "free surface," "Traction free", "stress free" all equivalent Neumann boundary conditions?
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166 views

Understanding Boundary Condition in FEM

I am trying to understand Dirichlet and Neumann boundary conditions in FEM and I wanted to know if my inference is correct. To articulate my understanding, lets consider a simple case of TE and TM ...
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2answers
66 views

How do you apply boundary conditions in a time-stepping problem?

It looks to me like a very common problem, yet I haven't been able to find any practical guide on the subject despite many hours searching. Here is a clearer statement of my question: I have a ...
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1answer
137 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 ...
3
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1answer
189 views

FEM current toy problem

I am solving the Dirichlet problem $$ \begin{cases} \Delta u = 0, \\ u|_{\partial D} = f, \end{cases} $$ in a $2d$ domain $D$ using the finite element method. What I want to get is the ...
5
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1answer
360 views

How to apply non zero Dirichlet boundary condition in finite elements?

I am writing a code for steady state heat transfer on a rectangular domain. I am specifying temperature on the edges - nonzero Dirichlet boundary condition. The equations can be written in form of $$...
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1answer
181 views

Neumann boundary conditions diffusion equations methods of lines

I want to solve the diffusion equation using the method of lines with Neumann boundary conditions $$ \frac{\partial p}{\partial t}=\frac{\partial^2p}{\partial x^2}\\ \frac{\partial p}{\partial x}(x=0)=...
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1answer
139 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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105 views

Do practice and theory differ substantially when implementing Neumann Boundary Conditions using a Mixed Method?

I have implemented a pretty straightforward finite element solver for the following Poisson equation. For the purposes of this question we can assume the source term and the Dirichlet data both ...