would like to compute a tensor derivative of a function and
in [Mathematica]
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From: nooj on 10 Dec 2009 04:56 <anonymous> wrote: > Hi, > you should explain what you exactly mean by "tensorial derivative". > > If you really mean the covariant derivative then you did not give enough > information. > > But maybe you mean the ordinary derivative. E.g.: D[ W[C[x,y,z]], x]. > Here we have the problem that Mathematica has no general rules for > the derivative of Det. Therefore, either you specify these rules or you > give the components of C explicitely as functions of x,y,z, so that > Det can be evaluated. > > Further, note that capital C is a reversed symbol, I changed it to lower > case. Custom names should always begin with lowercase, system > names always begin with a capital. > > For simplicity I make a 2D example: > > c[x_,y_]:= {{x+0.5 x^2,x},{x,1+2 x }}; > Wiso[c_] := a Exp[ b(I1[c]-3)^2 ] - a; > Wdil[c_] := k ( Det[c]^2 - 1 - Log[ Det[c] ] ); > W[c_] := Wiso[c] + Wdil[c]; > > Now we can get our derivatives as usual. E.g.: > D[W[c[x, y]], x] > D[W[c[x, y]], {x,2}] e.t.c Hi, I mean this derivative: http://en.wikipedia.org/wiki/Tensor_derivative_(continuum_mechanics) So the first derivative of W(C) with respect to C would be this kind of derivative: http://en.wikipedia.org/wiki/Tensor_derivative_(continuum_mechanics)#Derivatives_of_scalar_valued_functions_of_second-order_tensors And the first derivative of that would be http://en.wikipedia.org/wiki/Tensor_derivative_(continuum_mechanics)#Derivatives_of_tensor_valued_functions_of_second-order_tensors Good catch on the "C"; I was typing that in from memory. I'll use "A" from now on. I'll also ignore the part about how I was taking the difference with respect to a symmetric tensor; I can handle that distinction (See http://en.wikipedia.org/wiki/Tensor_derivative_(continuum_mechanics)#Derivative_of_a_second-order_tensor_with_respect_to_itself). The derivative of Det[A] can be easily stated (though it's not straightforward to compute): partial Det[A] / partial A = Det[A] A^(-T). (See http://en.wikipedia.org/wiki/Tensor_derivative_(continuum_mechanics)#Derivative_of_a_second-order_tensor_with_respect_to_itself) This is what I tried: /* components of the matrices */ Clear[a, b, c, d, e, f, g, h, i] /* nonsymmetric tensor */ A = ( { {a, b, c}, {d, e, f}, {g, h, i} } ) /* a constant symmetric tensor */ C0 = ( { {1, 0, 1}, {0, 2, 0}, {1, 0, 3} } ) /* this works, gives the identity tensor */ D[Tr[A], {A, 1}] // MatrixForm /* multiplier to build Cbar */ Jmod[A] := Det[A]^(-1/3) /* Cbar is like Ct but has determinant 1 */ Cbar[A_] := Jmod[A] A /* make sure Cbar looks okay */ Cbar[A] // MatrixForm /* This is my strain energy function. */ W[A_] := \[Alpha] / (2 \[Gamma]) ( Exp[\[Gamma] (Tr[Cbar[A]] - 3)^2] - 1 ) /* Is this what I want? How do I evaluate it at C0? */ D[W[A], {A, 1}] /* Is this what I want? It's fourth rank. How do I evaluate it? */ D[W[A], {A, 2}] - Nooj
From: Jeff on 12 Dec 2009 06:58 You might want to look at NCAlgebra, a package for non-commutative algebra that expands on the vector and matrix commands in Mathematica. http://math.ucsd.edu/~ncalg/
From: magma on 15 Dec 2009 07:25
If you are seriously working with tensors and you know what you are doing with them (meaning: you know differential geometry), you should try xAct (free package suite) or Tensorial (low-cost-high quality David Park's package with examples from continuous mechanics too) or TContinuumMechanics2 (a package by Jean-Fran=E7ois Gouyet that adds additional commands to Tensorial 4.0 for working in continuum mechanics) Worth mentioning is also the book: Continuum-Mechanics-using- Mathematica Links: XAct http://metric.iem.csic.es/Martin-Garcia/xAct/index.html Tensorial/TContinuumMechanics2 http://home.comcast.net/~djmpark/TensorialPage.html Continuum-Mechanics-using-Mathematica http://www.amazon.com/Continuum-Mechanics-using-Mathematica%C2%AE-Fundament= als/dp/0817632409 |