Can Triangulation and Pivoting Derive U and V for Pseudo-Arclength Continuation?

[allezz]

Homework Statement

[Dx De ; 0 0 ].[ U ; V ] = [ 0 0 ]

U : Nx1 vector
V : a scalar
Dx : NxN matrice ( dL / dx )
De : Nx1 vector ( dL / dE )

L is set of equations ( N amount )
E is a parameter in equations
x is unknowns ( N amount )


I need to derive U and V ( tangent vector components ) to apply pseudo-arclenght method to solve set of equations

I need to find a supplementary equation in form of

U(X-Xo) + V(E-Eo) - S = 0

S is the arclength step size

Homework Equations

Is it derivable by hand or do i need a algorithm to find U and V ?

The Attempt at a Solution

if i set V scalar as 1

U = inv(Dx)(-De) i can deduce U components

but if V is zero then i can't find N-dimensional vector ,

I heard that it can be done with triangulation and complete pivoting. How could it be possible?

 

[mighty2000]

it is possible to derive the values of U and V by hand using the given information. Here are the steps you can follow:

1. Rewrite the given equation as [Dx De; 0 0].[U; V] = [0; 0]. This is a matrix equation that can be written as two separate equations:

DxU + DeV = 0
0U + 0V = 0

2. Since the second equation is always satisfied, we can ignore it and focus on the first equation. Rearrange it to get:

V = -DxU/De

3. Now, we can substitute this value of V into the first equation to get:

DxU - DxU = 0

This means that U can take any value, as long as V is calculated using the above equation.

4. To find U and V, we need to also satisfy the supplementary equation given. Using the value of V we calculated in step 2, we can rewrite the supplementary equation as:

U(X-Xo) - DxU(E-Eo)/De - S = 0

5. This is a linear equation in terms of U, which can be solved using standard methods such as Gaussian elimination or LU decomposition. Once you have the value of U, you can use the equation in step 2 to calculate the corresponding value of V.

In general, it is possible to derive the values of U and V by hand using the given information. However, in some cases, it may be easier to use an algorithm or a software program to solve the equations. This could be the case if the equations are complex or if there are a large number of unknowns. In such cases, using triangulation and complete pivoting can be helpful in simplifying the equations and making them easier to solve. I hope this helps. Good luck with your calculations!