The kinetics of damage is given in differential form. The Differential equation is attached in jpg format(equation 81 of reference paper).In the equation take n=3,i(=1,2,3) do not denote index summation.S,k,C0 are known. The RHS of the differential equation is known.The damage in three orthogonal directions is to be found.Which numerical method I should follow to obtain D1,D2,D3.What are the reference books or papers to be followed to obtain the numerical solution of the given differential equation?
example:D1(i=1) is the damage variable corresponding to the principal stress1,
λt=modified growth defect density for each principal stress which obtained(equation 83 of reference paper).
I need to find out the damage corresponding to each principal stress direction.I use this damage values to update my compliance tensor.I am implementing the damage model in abaqus using user subroutines for which I need to write a code for solving to find the damage values in each time step for the above nonlinear differential equation in FORTRAN.
Reference paper:A probabilistic damage model of the dynamic fragmentation process in brittle materials,Pascal,Hild,Advances in applied mechanics,2010, 44, pp.1-72.
I assume your problem is an initial value problem. Apply Runge-Kutta method to find solution.
Can you first explain the notations or at least give a source for your equation ?
First reduce the equation to system of n ordinary equations and them you can apply Runge-Kutta or other similar method for initial value problem
MY SUGGESTION IS TO LOOK MATLAB ENVIRONMENT. MORE INFORMATION ABOUT THE EQUATION IS NEEDED, BUT AT THE BEGINING THE RUNGE-KUTTA METHOD IS A GOOD IDEA.
You can use the numerical method presented in paper:
A New Direct Time Integration Method for the Equations of Motion in Structural Dynamics
ZAMM · Z. Angew. Math. Mech. 94, No. 9, 757 – 774 (2014) / DOI 10.1002/zamm.201200245, by J.T. Katsikadelis.
After modifying it appropriately for higher order equations ,
Apart from various methods and softwares developed for solving initial value problems (IVPs), such as your problem, the Matlab function ode45 which is based on an explicit Runge-Kutta (4, 5) formula, is a proper solver for your problem. This solver controls the error by two parameters RelTol and AbsTol. By using these parameters, you can also adjust the relative error tolerance and the absolute error tolerances.
Have you try the FLEXPDE? It is easy and allow you to do nice analyses. There are a student version that i think it is enough to your problem.
Hello ,
Sir Vladimir Rasvan.I have edited the the question and given the explanation of the terms involved and also given the source of the equation.
Bhanu, A quick look at the equation tells me that the DEs are not coupled and hence can be solved independently. Since the RHS of the DEs ae independent functions, the following suggestions might help in solving the DEs.
1. Solve the first n-1 derivatives (just n-1 times integrations of RHS) to get
1/(1-Di) dDi/dt = F(t).
2. Multiply by (1-Di) to get dDi/dt + F(t)Di = F(t)
3. Solve this first order LINEAR ODE by method of integrating factor.
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