Let me correct your question to the following one: how could I apprioximate a nonlinear term by a linear one? Detailed answer must be based on formulation of your problem.
1) For instance, if your problem concerns a nonlinear partial differential equation (NPDE) of the following form (H+eV)f=r, where H is linear operator, V is your non-linear term, e is perturbation small parameter and f is your target then you can apply perturbation method.
2) There are nonlinear problems which can be solved exactly. For instance there is class 1+1 dimensional of NPDE which can be handled by Inverse scattering method, or by the Backlund transformation. Et cetera. Display more about your problem.
Let me correct your question to the following one: how could I apprioximate a nonlinear term by a linear one? Detailed answer must be based on formulation of your problem.
1) For instance, if your problem concerns a nonlinear partial differential equation (NPDE) of the following form (H+eV)f=r, where H is linear operator, V is your non-linear term, e is perturbation small parameter and f is your target then you can apply perturbation method.
2) There are nonlinear problems which can be solved exactly. For instance there is class 1+1 dimensional of NPDE which can be handled by Inverse scattering method, or by the Backlund transformation. Et cetera. Display more about your problem.
You may also use quasi-linearization. Suppose your non-linear term is f(u, ux) etc and initial values of u and ux are given. Suppose these values are u(0) and ux(0).Then f(u,ux) can be approximated as
f(u, ux) = f(u(0), ux(0)) + (u - u(0)) fu (0)+ (ux -ux(0))fux(0)
If you have higher order derivatives , one can use that higher order derivative in the approximation.If ux or u is not there that term will be dropped.
So if we approximate nonlinear term by a linear one,why we choose meta heuristic to find an optimal solution in some cases?
Still you do't reveal details of your problem and our discussion becomes meaning less.
Depending on the type of nonlinearity of the equations- eg. exponential, power or saturation-growth-rate, you can linearize using ln, log or inverse functions to linearise the relationship. Book like "Numerical Method with applications" By Chapra is a good guide.
you can approximate result by eliminating the non linear terms. different techniques have been adopted to eliminate the non linear terms or approximate to linear. these techniques also named as numerical analysis technique. my favorite technique is method of multiple scale developed by Prof Ali Nayfeh of virginia tech
First you must identify your problem clearly so we can answer it correctly.
1.for finite element analysis ignore geometric and material non linearity.
2. For mathematical problems approximate quadratic functions to linear ones.
3.for geometric problems indeterminate structures can be approximated to determinate ones and then increase the obtained results as stresses .deflections. forces by 25%.
If your term involves an arbitrary (Frechet) differentiable mapping f defined on an open subset of a normed vector space, then the expression f(a+h) - f(a) can be approximated by f ' (a)(h), (which is linear in h), but only for Norm(h) sufficiently small.
Hello
Linearization technique depends on the type of problem. In finite element applications quasi linerization is achieved by an iteration technique whereby the algebraic sytsem is solved two or more times until the solution converges.
Consider the solution of the non-linear PDE u*uxx +uyy + k = 0 by FEM.
The 1st term in nonlinear. The PDE is solved by temporarily assuming that u in u*uxx is a constant say uo. Then the PDE is solved by assuming reasonable values for vector uo in the 1st iteration to yield a new set of values for uo. This new set of values is then used in the 2nd iteration to yield another set of values for uo. This process shall continue until the vaues of uo converge.
The FEM equivalent of the PDE is
∫ uo*BxT*Bx + ByT*By + K*NT dxdy = 0
where Bx = ∂N/ ∂x , By = ∂N/∂y , and N = shape function matrix
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