Questions tagged [numerical-methods]
Questions concerning problems which cannot be exactly solved, or where the exact solution may be much more difficult to acquire than a solution by numerical methods.
48
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Casting and the pressure of the metal
Background
A gold casting process is explained here:
The casting metal is melted in a special casting machine that is programmed to insert the metal under vacuum into the flask at the right time, ...
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Accuracy of stable (not backward stable) numerical algorithm
I understand the definition of accuracy for a backward stable numerical algorithm, which shows how such a quantity depends both on machine-epsilon and on the condition number of the problem under ...
- 151
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1
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62
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Implementing convective boundary conditions for 1-dimensional finite difference computation of heat equation
I am simulating heat transfer through a 1-dimensional wall with convection on both sides. For the wall, I implemented a finite difference model like this:
$$
u(x, t + ∆t) ≈ u(x, t) + c [u(x + ∆x, t) − ...
- 3
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1
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Inverse analysis of one Storey structure
The displacement history of one story structure is known due to an unknown ground motion. The dynamic properties, such as mass and lateral stiffness, are known, and dimensions were also given. How to ...
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2
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274
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How to develop the the Backward Euler method for a State Space
I have been trying to develop to solve a state space model in discrete time.
I have the following equation for my state space:
$$\dot{x} = Ax + Bu, $$
I am developping the equation, but I do not ...
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Structured trianglar finite element mesh in MATALB
The available triangular mesh in the MATLAB PDE tool box here gives an unstructured triangular mesh.
How can I get a structured triangular mesh from the tool box with a similar data structure [p,e,t]. ...
- 101
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In FEM, can each instance of a Transient Analysis be thought of as a Static Analysis of its own?
Assume I have a cantilever beam fixed at one end. I can conduct a static analysis in FEM for a case where I apply a load on the other end, condition being the load is consistently applied on it and ...
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In FEM, what is the difference between a single element with a quadratic shape function and two elements with linear shape functions?
Using Finite Element Analysis to obtain a Weak form of a PDE, what is the difference between the two cases:
A single element with a quadratic shape function
Two elements with linear shape functions.
...
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2
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Why does it take lesser iterations for a stiffer model in my FEA than a less stiffer model to converge?
I am currently using ANSYS FEA software and I conducted a simple geometric non-linear analysis of two same shaped bodies, but different cross sectional areas; one being quite bigger than the other. We ...
- 1,337
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Does numerical dissipation in FE simulations affect system's elastic energy? Does it depend on thickness?
I am running a finite-element simulation in Abaqus, for which I also have a theoretical model. To be precise, I am simulating wrinkling in a thin annular sheet, with shell elements. The model makes a ...
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Why is the amount of force/moment transfer occurring between bodies in FEA is independent of mesh size?
So I read somewhere that the mesh size won't effect the amount of force transfer occurring between bodies if several bodies are in contact, and a global force is applied somewhere. I wanted to know ...
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2
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What is the difference between Frictional non-linearity and Geometric non-linearity in FEA? (In terms of Stiffness matrix)
For geometric non-linearity, we know that as the geometry changes during the analysis, the global stiffness matrix changes since it already is a function of geometry of the structure.
What I want to ...
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Errors obtained in the numerical solution of the 1D convection-diffusion equation
I have been trying to solve the 1D convection-diffusion equation
In implementing the above, I have tried to follow closely the method described here: https://uk.mathworks.com/support/search.html/...
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Analytical solution for the 1D convection-diffusion equation
The Ogata and Banks analytical solution of the convection-diffusion equation for a continuous source of infinite duration and a 1D domain:
where C [mol/L] is the ...
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2
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176
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How to convert the velocity field to a pressure field for obtaining the sound level?
I am studying jet mixing noise and I am given a velocity field, with $V_x$, $V_y$, $V_z$ in the Cartesian coordinates. I wanted to know if there is a method by which I could find some information ...
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1
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Finite difference discretization of the Cauchy-Riemann PDEs
I made a forward fd-discretization of the Cauchy-Riemann PDEs but I am struggling to implement this in python.
I have a quadratic mesh with heighτ = $2*\pi$. The dirichlet boundary conditions are at $...
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Convergence problem in RANS simulation of unsteady flow
I am simulating an air flow through a two dimensional rectangular channel. The channel splits into two channels, one of them is followed by a 90 degree bend downwards and leads into a cylindrical ...
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1
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Extracting Uncertainty from Numerical Solution
I am performing numerical and uncertainty analysis for an oblique shockwave angle function:
$$
\tan(\delta)=\frac{2}{\tan(\theta)}\frac{M^2\sin^2(\theta)-1}{M^2(\gamma+\cos(2\theta))+2}
$$
where $\...
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0
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Is it possible to measure the concrete workability by a mobile app one day? [closed]
After studying all methods that can measure the concrete slump, I found 62 devices that service this purpose at the same time these devices are expensive and not practical enough, for this reason the ...
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Does Finite Element Method use direct stiffness method?
Does Finite Element Method use direct stiffness method? Or it is another mathematical method? We definitely use direct stiffness method to solve frames. FEM is based on entirely this or depending on ...
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Analytical solution of thermal stress in a bi-metallic cylinder
As a side project I have looked to solve using Python an analytical solution for one-dimensional transient heat transfer in a thick cylinder with cladding on the inside surface, wherein the fluid in ...
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Water distribution through N x Needle orifices
Consider a water distribution setup consisting of a feeder line blocked at one end but having 100 needle outlets all the same size. The feeder line cross sectional area is much greater than a needle ...
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How to use the Ritz method with the weak form to approximate solution of differential equation
I am having trouble understanding how to approximate the solution to this problem using the Ritz method and the weak form:
$$\frac{d^2u}{dx^2} - u=0; \ \ x \in [0,1]$$
$$u(x=0)=0; \ \ \frac{du}{dx} \...
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Defining appropriate test function spaces for the finite element solution of Euler's fluid equations
I have the following coupled equations for the conservation of mass and momentum of a compressible fluid
\begin{equation}
\rho_t + (\rho u)_z = 0,
\end{equation}
$$
(\rho u)_t + (\rho u^2)_z + \...
- 101
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votes
1
answer
60
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Negative Eigenvalues in Ritz Solution of First-Order Shear Deformation Theory
I am trying to write a MATLAb code for determining the natural frequencies of a first-order shear deformation plate using Ritz method. I am using Legendre polynomials as trial functions in Ritz method,...
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Simufact / MARC and Fortran
I am using Simufact from MSC Software to perform numerical simulations.
To be able to create my own sub-routines I need to have Fortran compiler installed.
Simufact manual mentions only Intel Fortran +...
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1
answer
138
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How to code this iterative scheme for solving 2-group diffusion equation?
I am trying to write a python code to solve the neutron diffusion equation to model neutron flux distribution in a one-dimensional two-group setting. The governing equations of the system are:
$$-...
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answer
265
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Implicit and Explicit finite difference discretization of steady state heat conduction equation
I understand what an implicit and explicit form of finite-difference (FD) discretization for the transient heat conduction equation means. But I am not able to understand if it is possible to ...
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364
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FEA: Newton-Raphson algorithm for Dirichlet BC non-linear static analysis
I have a standard non-linear static problem to solve with FEA. I am applying displacement boundary conditions only, no external force. The equation to solve look like these:
$K(u)u=R(u)$ Eq(1)
The ...
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134
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Calculate surface area and outward normal of each face(non-planar) for a irregular hexahedron.
I am using the vector cross product to calculate each surface area and thus the outward normal. However, this approach fails when the surfaces are non-planar. So what is the best approach to calculate ...
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Temperature not changing in my code for 1-D heat equation: Explicit Scheme
I am trying to solve the following 1-D heat equation with provided boundary conditions using explicit scheme on Matlab. I have been trying to plot the results but I realized that my temperatures are ...
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2
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Example of real-life problem solved with numerical methods? [closed]
I hope this is a suitable question to ask here.
Well, I'm taking Numerical Methods this semester (also called 'Numerical Analysis' in some places).
The term-assignment is to find a real-life problem ...
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How to formulate a 3D version of the Navier-Stokes equations in 2D for numerical solution?
I'm considering a topology optimization problem where the design domain is a rotating annulus (in $\mathbb{R}^2$). One of the adjoint equations of the Navier-Stokes system whose solution is to be used ...
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How does Abaqus calculate reaction force?
I wrote a UEL for one element and compared to a reference UEL (FEM), where the results agreed satisfactorily, except the reaction force. The stress, strain, strain energy all were pretty accurate.
I ...
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Discretise this transport equation
I am trying to discretise the PDE: $$\phi \frac{\partial c}{\partial t}+\frac{\partial j}{\partial x}=0$$ where $c$ is a function of $x$ and $t$, and $j=qc-D(q)\frac{\partial c}{\partial x}$, $q$ is ...
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468
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How to derive deformation gradient or displacement gradient?
In one of my project, I need deformation gradient or displacement gradient for some elements in my model that made by ABAQUS software and I can't find it directly, any suggestion or help would be ...
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Mechanical Engineering- Modeling contact in Abaqus
I have simulated a 2D model in Abaqus which indicates 2 main part, first a sector that represent as wheel and, second a rectangular that represent rail. the wheel and rail are in contact and wheel ...
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Does excitation matter in a time-domain simulation/computation of a transfer function of an LTI system?
Let's say I am running a FDTD simulation of a wave-equation to determine a transfer function of an LTI system:
\begin{equation}
H(f) = \dfrac{Y(f)}{X(f)}\
\end{equation}
where $Y(f)$ and $X(f)$ are ...
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4
answers
3k
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J2 orbit propagation problem in MATLAB (solver instability)
The problem is that when I try to solve the following equations of J2 perturbed gravity:
...
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answers
2k
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How to calculate effective strain in plane stress?
What is the correct formula to calculate the effective strain under plane stress conditions?
In plane stress it is assumed that the out of plane stresses are zero. The out of plane strains are not ...
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Deflection of thin plate with 1 free edge and deflection > thickness
I have an thin uniform rectangular steel plate supported on 3 sides and free on one, and supporting a uniform load.
I'm trying to determine what thickness of plate I need to keep deflection below a ...
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Normalization and Non-Dimensionalization of a mass balance equation
I would like to ask what is the difference (if any) between normalization and non-dimensionalization. I will try to explain what I have done until now with an equation that I would like to work with ...
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Sliding mode surface derivative exact matrix format (ds/dx)
Is there any suggestion on sliding surface state derivative matrix (ds/dx) format?
...
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Sliding mode surface derivative using numerical method
In the article Controller Design using Bifurcation Map for Aircraft Spin Recovery, the sliding surface derivative is obtained using a numerical method.
The steady space system is defined by the ...
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2
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External diffusion: calculation of surface concentration
I am struggling a bit with an external diffusion problem. I am trying to calculate the concentration at the surface (as well as surface reaction rate) and would like some help or guidance.
Here is ...
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Verify parameters variation in local sensitivity coefficients for microkinetic model
I’m performing a sensitivity analysis for a model of a set of elementary chemical reactions. I’m considering the variation in species concentration ($c_{k}$) with respect to the forward reaction rate ...
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458
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Different calculations for values of thermodynamic properties
While deciding what tool (library) to use for a simulation of a hybrid gas turbine plant, I found that different libraries/tools of thermodynamic properties yield different values than the other.
I ...
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How are thermodynamics properties of gases and liquids calculated numerically?
I was wondering how the values of thermodynamic properties as enthalpy, entropy, internal energy, specific heat, specific volumes ... etc, of gases and liquids are calculated using programming ...
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