CODEWITHSUNDEEP
python-3.xnumpymatrixlinear-algebra
Jek Denys
Jek Denys

Reputation: 115

Solving matrix equations with some Boundary Conditions

How can I solve a system of linear equations with some Boundary conditions, using Numpy?

Ax=B

Where x is a column vector with, let's say x1=0. For different iterations BCs are going to be different, so different variables of vector x going to be zero. [A] and [B] are known.

Here is an example from my FEM course:
{F} Is the column vector of known values
[k] is the stiffness matrix with the known values
{U} is the displacement column vector where U1 and U3 are known to be zero, but U2 and U4 need to be found.

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Here is an example:

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This would result in these values:

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Naturally this would reduce to the 2X2 matrix equation, but I because for different elements the BC would be different, I'm looking for some numpy matrix equation solver where I can let it know that some of the unknowns must be this certain value and nothing else.
Is there something similar to np.linalg.solve() with conditions to it?

Thank you.

Upvotes: 2

Views: 1890

Answers (1)

hiro protagonist
hiro protagonist

Reputation: 46921

the matrix k in your example is invertible. that means there is one and only one solution; you can not choose any of the Us. this is the solution:

import numpy as np

k = np.array(((1000, 0, -1000, 0),
              (0, 3000, 0, -3000),
              (-100, 0, 3000, -2000),
              (0, -3000, -2000, 5000)))

F = np.array((0, 0, 0, 5000))

U = np.linalg.solve(k, F)
print(U)
# # or:
# k_inv = np.linalg.inv(k)
# U = k_inv.dot(F)

# [ 5.55555556  8.05555556  5.55555556  8.05555556]

the same in sage:

k = matrix(((1000, 0, -1000, 0),
              (0, 3000, 0, -3000),
              (-100, 0, 3000, -2000),
              (0, -3000, -2000, 5000)))
F = vector((0, 0, 0, 5000))
U = k.inverse() * F

# (50/9, 145/18, 50/9, 145/18)

Upvotes: 1

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