Hi everyone, I am trying to implement a thermo-mechanical coupled constitutive model of bulk metallic glass into ABAQUS by VUMAT. The rate of temperature change is a fuction of current temperature and rate of plastic strain in the material with mathematic expression

dT/dt = A Laplacian(T) + B dP/dt

Where T is temperature, Laplacian(T) = d^2(T)/(dx)^2 + d^2(T)/(dy)^2, A and B are material coefficient, and dP/dt is plastic strain rate. In other words, the temperature change of the metallic glass are caused by two mechanisms, one is diffusion of heat (Laplacian), and another is heat generation due to plastic dissipation. In order to implement it into VUMAT, I convert the PDE above into forward Euler incremental equation as

dT= A(dt) Laplacian(T) + B dP

So far I manage to determine value of A, B and dP. If I could find the value of Laplacian(T) on each material point, I can solve the 2nd equation to determine value of dT as dt will be provided by ABAQUS. Then, I can use the dT value to update temperature at each material point by equation

TempNew = TempOld + dt

However, my problem is I don’t know how to implement the Laplacian into VUMAT. It is because by using finite difference method, Laplacian of a function takes the form as

Laplacian ( f(x,y) ) = ( f(x-h,y) + f(x+h,y) + f(x,y-h) + f(x,y+h) - 4f(x,y) ) / h^2

which means in order to determine value of Laplacian(T) in a material point, I need to call back temperature value of 4 nearby material points, and the h value, which is the distance between the material point I want to study and its neighbor material point. Is anyone has experience in solving this kind of problems? Or my problem can simply be solved by using some built in function in ABAQUS? Any opinion or suggestion on how to dealing with Laplacian term using VUMAT are welcomed, and that would be great if can provide the example solution for this kind of problem.

Thank you.

P.S. This problem is actually identical with the problem discussed in blog below, and applogize to Zhang and Alexander as didn't reply on time.

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Hey everyone, I am trying to implement constitutive equations of bulk metallic glass into ABAQUS by VUMAT. However, the free volume concentration equation in the constitutive equations consists a diffusion terms which required to solve a Laplacian operation. The simplified free volume concentration equation is shown below:

dV/dt = A Laplacian(V) + B dP/dt

Where V is free volume concentration, Laplacian(V) = d^2(V)/(dx)^2 + d^2(V)/(dy)^2, A and B are material coefficient, and dP/dt is plastic strain rate. In order to implement it into VUMAT, I convert the PDE above into forward Euler incremental equation as below

dV= A(dt) Laplacian(V) + B dP

I understand that by the finite difference method, Laplacian takes the form as

Laplacian ( f(x,y) ) = ( f(x-h,y) + f(x+h,y) + f(x,y-h) + f(x,y+h) - 4f(x,y) ) / h^2

However, I still don’t know how to implement the Laplacian into VUMAT because according to finite Laplacian equation above, in order to obtain dV in a material point, I need to call back free volume concentration value of 4 nearby material points, and the h value, which is the distance between the material point I want to study and its neighbor material point. Is anyone has experience in solving this kind of problems? Or my problem can simply be solved by using some built in function in ABAQUS? Any opinion or suggestion are welcomed, and that would be great if can provide the example solution for this kind of problem. Thanks and regards.

I am working on some boundary integral equation formulation and I am currently stuck with some mathematics. I wish anyone can help me out with this.

I have an anisotropic (sometimes called generalized) biharmonic differential operator which takes the form

L = k11 D1^4 + k12 D1^2 D2^2 + k22 D2^4

where D1 = d/dx, D2 = d/dy, my problem is two dimensional.

I need to find a fundamental solution (Green's function) for this operator, that is

L(u) = -delta

where delta is Dirac delta function.

I thought of two solutions but they both seem to fail, first, i thought of making some coordinate transformation so that the coefficients of the terms of the operator become the same, and hence, use the traditional biharmonic Green's function (1/8/pi r^2 ln(r) ), however, my proposed transformation takes the form

xi = a11 x + a12 y

eta = a21 x + a22 y

there a11, a12, a21 and a22 are to be determined according to the above requirement (all operator terms have equal coefficients), however, using this transformation got me more terms in the final operator form (terms involving D1^3 D2 and D1 D2^3)

My second solution was to decompose the anisotropic biharmonic operator into two anisotropic laplacian operators and solving two anisotropic laplacian equations instead of one, I succeeded in the decomposition and I've solved the first anisotropic laplacian equation, however, I cannot solve the seconf one. the solution I got was the following

L = (a1 D1^2 + a2 D2^2)*(b1 D1^2 + b2 D2^2)

let L1 = (a1 D1^2 + a2 D2^2)

and

L2 = (b1 D1^2 + b2 D2^2)

now, L1(L2(u)) = -delta

let L2(u) = v

then L1(v) = -delta

this gives v = -1/sqrt(a1*a2)*ln(r')

where r' = sqrt(x^2/a1 + y^2/a2)

now, we have L2(u) = v

thus L2(u) = -1/2/sqrt(a1*a2)*ln(x^2/a1 + y^2/a2)

which gives (b1 D1^2 + b2 D2^2)(u) = -1/2/sqrt(a1*a2)*ln(x^2/a1 + y^2/a2)

I am stuck at this point, I have no clue how to solve this

any ideas ??

thanks

Ahmed