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The answer is using a thermodynamic equation of state (EOS). An EOS is one that relates the free energy of a substance to its physical properties (temperature and density for example). Once such an equation is developed all other thermodynamic properties can be derived from them (via thermodynamic relationships). To see an example for water, checkout the The ...

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It's hard to say exactly which method they would use with certainty, but it's pretty reasonable to assume that they'd specify the method if it was anything special. They easiest way to do this would be with a finite difference scheme. In this you approximate a derivative of a function by computing it at two locations that are a small, but known distance ...

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You are correct that this problem falls within the category of 'large-deflection' problems since the deflection is larger then about $\frac{t}{2}$. To correctly analyse a plate under large-deflections would require solving the Föppl–von Kármán equations, which is generally not possible except for some very specific cases. Numerical solutions to some cases ...

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From a theoretical standpoint, the displacement gradient is equivalent to strain (assuming a structural problem). Numerically, you can obtain the derivate of a quantity through multiplication with the derivative of the shape functions, which is often referred to as the B matrix: du = B u

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The only way you can get negative eigenvalues is by including stress stiffness effects (sometimes called "geometric stiffness"), if there are compressive stresses which would cause the plate to buckle. There can be zero eigenvalues if the plate can move as a rigid body, of course, and they might be calculated as small negative numbers, but those should be a ...

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The discretized problem that you are trying to solve is $$\text{find}\,\, \mathbf{u}^{n+1} \,\, \text{such that} \,\, \mathbf{r}(\mathbf{u}^{n+1}) = \mathbf{0} \,\, \text{subject to the constraints} \,\, \mathbf{g}(\mathbf{u}^{n+1}) = \mathbf{0}$$ where $\mathbf{u}^{n+1}$ is the displacement at the end of load step $n$ and the residual is $$\mathbf{r}(\... 2 You could develop some saturated steam equations by running a regression on steam table data. Even if it was a giant equation, sure would be nice to not have to do a a table lookup in an otherwise automatic spreadsheet or script. I attempted to use some I found on the internet the other day and only one of them worked (saturated steam equations SE question). ... 2 FE solvers are not dependent on the direct stiffness method. You can introduce conceptually the FE methodology to students using the direct stiffness method. FEM Solvers use different formulations/schemes. For example, you'd probably use different approaches for a explicit or an implicit solver. Or you might need to use different formulations depending on ... 2 Everything in your OP looks correct. I think you have just got into a muddle with the notation, and what is known and what is unknown. \phi_1 and \phi_2 are known functions, i.e. you choose two sensible functions for the problem you want to solve, to give a "good" approximation to the solution you expect. You can differentiate them with respect to x.... 2 depending on the type of non linear analysis you used, then conceptually the analysis takes the following steps (again depending on the type on analysis you use): An iterative process starts: A small increment of the load \Delta F is applied (typically a percentage between 1% and 10%) For those small load increments the displacements \delta_i are ... 1 I'll try to address the question in bold. The expression for the angle function can be expressed as f(\delta, M; \theta). You can compute partial derivatives of this function as$$ \frac{\partial f}{\partial \delta} = \frac{\partial f}{\partial \theta} \frac{\partial \theta}{\partial \delta} $$and$$ \frac{\partial f}{\partial M} = \frac{\partial f}{...

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You haven't given enough information to answer this exactly, but I am assuming from your equations that you are solving mesh-centered equations and the assumption is that the flux is linear in each cell. In that case, the average value of the flux in each cell is taken as the midpoint, and your integration just becomes a sum $$k^n=k^{n-1}\frac{\sum h(\nu_1\... 1 A non-iterative solution for steady-state heat flow must be implicit. You are solving for the temperatures "everywhere" in the structure in a single "step", so every boundary condition must affect the entire temperature distribution in the equations you are solving. Iterative solutions for steady-state conditions can be either implicit or explicit. You can ... 1 If the surface is nonplanar then the surface normal is a (hopefully continuous and differentiable) function of location. You will have to integrate the surface function S(x,y,z) over any two of the dimensions. If this is an engineering problem, then of course use a finite-grid approach and numerical integration. Then treat each grid element as a planar ... 1 This stackexchange doesn't really seek to solve these homework-like problems directly. However, I can offer some hints on how to proceed. Are you sure the scale of a and lambda are correct? When I assign a=0.1, a reasonable profile develops. Consider non-dimensionalizing your problem so the variables all scale from 0 to 1. These kinds of errors tend to ... 1 A heat transfer problem might be suitable here. Temperature is an intuitive parameter and a simple scalar, and energy balances are easy to write down but often challenging to solve analytically because of temperature-dependent material properties, for example. I wrote here about the parabolic and catenary temperature profiles that arise in a simple 1-D ... 1 There can be thousands of causes for your error. You need to give many more informations to be able to make a guess. Like: discretization, solving methods for the DEQs, grid topology, boundary conditions etc. 1 https://www.mathworks.com/matlabcentral/fileexchange/55167-high-precision-orbit-propagator?s_tid=srchtitle https://www.researchgate.net/publication/340793133_High_Precision_Orbit_Propagator_C_code The motion of a near-Earth satellite is affected by various forces. One of these forces is the Earth's central gravitation and the others are known as ... 1 Solved it, thanks to a colleague. The problem is, I used "J2 Propagator" in STK, which in fact uses secular rates of orbital precession. Comparing that result with the real integration that MATLAB does, led to this result. When I used HPOP propagator with degree 2 and order 0 (which is the same as just using J2 perturbation), the results got much better. ... 1 As far as I can see, you are trying to integrate a nonlinear ODE of a conservative system. It is very likely that your system exhibits chaotic behavior. RK4 is a fourth order numerical scheme. These numerical integrators tend to change the total energy of the system, which contradicts the conservativeness of the system but explains the differences between ... 1 The small deflection approximation is not based on deflection vs thickness, it is based on deflection over span. So for your maximum case of 10mm deflection on a 1250mm span, you are only looking at a 0.8% deflection. Another way to look at is from a statics point of view. Where the Deflection is one side of a triangle and half the span in the hypotenuse. ... 1 Sorry with not being familiar with all of your variables. If it is a mass balance as you say, then you can't get a Reynolds or Péclet number. A Reynolds number appears in the dimenional analysis of the impulse balance (the Reynolds number contains viscosity which originates from the stress tensor for Newtonian fluids). I'm also not very familiar with Péclet, ... 1 The way you have solved your problem you have treated the concentration at the surface of the sphere as known (y_{B,\text{surf}}). Notice that in your final answer, if you plug in r=r_\text{sphere} all you will get is y_{B,\text{surf}}. Instead, your boundary condition at the surface should be something like this:$$ N_{B,r}=-K_1P_B^{0.5}=-K_1y_B^{0.5}...

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What do you think is the reason for the difference in values? It's likely that they have calculated the coefficients of whatever model they are using from different data sets (or sub-sets). It's also possible that they are using different models. is this level of accuracy sufficient? That depends on what you're using it for. In general you probably have ...

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