Martin Fouilleul
3 years ago
parent
2767a75a1b
commit
a23167c57d
5 changed files with 151 additions and 14 deletions
@ -0,0 +1,106 @@ |
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from sympy import sqrt, symbols, factorial, IndexedBase, Symbol |
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p = IndexedBase("p") |
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h = Symbol("h") |
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def taylor(a, b, c): |
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n = 4 |
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derivatives = [0] * n |
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steps = [0] * n |
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result = p |
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for order in range(1, n+1): |
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for i in range(0, 3**order): |
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for j in range(0, order): |
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derivatives[j] = int((i / (3**(order-1-j))) % 3 + 1) |
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step = [a, b, c][derivatives[j] - 1] |
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steps[j] = step |
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total_step = 1 |
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symbol_name = "" |
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for j in range(0, order): |
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total_step *= (steps[j] * h) |
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component = derivatives[j] - 1 |
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symbol_name += ["x", "y", "z"][component] |
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result += (1 / factorial(order)) * total_step * Symbol(symbol_name) |
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return result; |
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def expand_laplacian(face, corner): |
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# Note: Any linear combination of the face and corner coefficients will |
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# produce a valid discretized Laplacian, except for when normalization == 0 |
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center = face * 4 + corner * 4 |
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normalization = face + 2 * corner |
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expr = ( |
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- center * taylor(0, 0, 0) |
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+ face * taylor( 1, 0, 0) |
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+ face * taylor(-1, 0, 0) |
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+ face * taylor( 0, -1, 0) |
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+ face * taylor( 0, 1, 0) |
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+ corner * taylor( 1, 1, 0) |
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+ corner * taylor( 1, -1, 0) |
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+ corner * taylor(-1, 1, 0) |
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+ corner * taylor(-1, -1, 0) |
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) / (normalization*h*h) |
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print("Laplacian:") |
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print("\tCenter: {:+d}".format(-center)) |
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print("\tFace: {:+d}".format(face)) |
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print("\tCorner: {:+d}".format(corner)) |
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print("\tNormalization: {:+d}".format(normalization)) |
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print("\tExpansion: {}".format(expr.simplify())) |
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print() |
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return expr |
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#a = expand_laplacian(1, 0) |
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#b = expand_laplacian(0, 1) |
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#c = expand_laplacian(1, 1) |
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#d = expand_laplacian(2, 1) |
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#e = expand_laplacian(1, 2) |
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expr_colocated = ( |
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-4*taylor(0, 0, 0) |
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+1*taylor(-2, 0, 0) |
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+1*taylor(+2, 0, 0) |
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+1*taylor(0, -2, 0) |
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+1*taylor(0, +2, 0) |
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) / (4*h*h) |
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expr_wide = ( |
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-8*taylor(0, 0, 0) |
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+1*taylor(-1, 0, 0) |
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+1*taylor(+1, 0, 0) |
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+1*taylor(0, -1, 0) |
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+1*taylor(0, +1, 0) |
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+1*taylor(-1,-1, 0) |
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+1*taylor(+1,+1, 0) |
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+1*taylor(+1, -1, 0) |
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+1*taylor(-1, +1, 0) |
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) / (3*h*h) |
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expr_vertex = ( |
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-4*taylor(0, 0, 0) |
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+1*taylor(-1,-1, 0) |
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+1*taylor(+1,+1, 0) |
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+1*taylor(+1, -1, 0) |
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+1*taylor(-1, +1, 0) |
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) / (2*h*h) |
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expr_mac = ( |
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-4*taylor(0, 0, 0) |
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+1*taylor(-1, 0, 0) |
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+1*taylor(+1, 0, 0) |
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+1*taylor(0, -1, 0) |
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+1*taylor(0, +1, 0) |
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) / (h*h) |
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print((expr_colocated - expr_mac).simplify()) |
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print((expr_colocated - expr_wide).simplify()) |
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print((expr_colocated - expr_vertex).simplify()) |
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@ -0,0 +1,14 @@ |
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#version 300 es |
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precision highp float; |
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in vec2 pos; |
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out vec2 texCoord; |
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uniform mat4 mvp; |
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void main() |
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{ |
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texCoord = 0.5*(pos + vec2(1,1)); |
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gl_Position = mvp * vec4(pos, 0, 1); |
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} |
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