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# Mathematical model and computer simulation

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1. What are mathematical models and computer simulations?
2. What are stochastic models and deterministic models?
3. What is convergence and numerical instability?
4. What are sensitivity analyses and uncertainty analyses?

https://brainmass.com/math/consumer-mathematics/mathematical-model-computer-simulation-375014

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1. A mathematical model is an equation, or set of equations, that describes a system by taking into account inputs, outputs, and internal states (possibly friction, heat generation, etc). A good example of this is saying that a spring-mass system (or KM system) could have a mathematical model that describes the natural frequency as omega=(spring stiffness/mass)^(1/2).

A computer simulation on the other hand is a program (either commercial or hand coded such as MATLAB) that is used to simulate a system. This could be useful if you need an estimate of a model that can not be solved in closed form. You could use iterations that will close in on the solution to the model; but stop iterations at a given percent of change. This is how you can setup some FEA models to simulate a system and then give the approximate solution once the results stop changing by a given percent (say maybe 0.001% or whatever you want to define).

2. A stochastic model is used to model a system that behaves randomly and could have a wide range of outputs for a range of inputs. The stochastic model will estimate the likelihood of each input/output relationship as a probability.

A deterministic model on the other hand is used to model a system and fix the values for the variables. This will give the same output value with the same input value every time and it will not give input/output relationships with a percentage like stochastic models; it will just be the same output each time for that given input.

3. Convergence instability can be seen most easily in FEA programs. The way that an FEA program runs is that it will make extremely large mass, stiffness, and damper matrices (could be millions by millions if the problem is complex). Many times when you get into very complicated simulations you will run into convergence instability and the problem either will not solve or you will have to take a convergence value that is higher (like maybe 1% instead of 0.01%).

Numerical instability means that when you implement an algorithm, round off, or truncation errors could cause significantly incorrect answers.

4. Sensitivity analysis is used to investigate the effect that a change in the input to a system will have on the output. A good example of this is natural frequency. If you are testing a tuning fork and you excite it with the frequency of the tuning fork then you will see the output change drastically with that input.

Uncertainty analysis, in terms of modeling, will provide a relative magnitude of the reliability of the model simulation when accounting for uncertainty in the model design as well as uncertainty in the model inputs.

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