User Defined Functions

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Some commands let you set properties using a function that is evaluated at run time.

Function Variables

When a user-defined function option is allowed in any command, you can enter any valid function of the following variables when doing MPM simulations:

For FEA calculations, the following variables are allowed and may refer to position of element centroid or node depending on the command:

When a user-defined function is used, it will be calculated using these variables and should return a result in the units expected by the command. Note that commands that allow functions may only allow a subset of these variables (due to command context). You can refer to each command for the allowed variables. For example, some MPM options (as detailed in their documentation) require the function to depend only on time. Particle-based MPM boundary conditions let the function depend on clockwise particle rotational angle q about the z axis (in radians), which allows rotation of the boundary conditions with the particle. Note that q is particle rotation since the start of the simulation and will differ from material angle if the particle started with a non-zero orientation angle (i.e., the current material angle is the sum of q and its initial angle).

When setting up MPM simulations or FEA calculations, a few more variables are sometimes allowed:

  • R - radial position in length units for axisymmetric calculation; R is a synonym for x, which also works
  • Z - axial position in length units for axisymmetric calculation; Z is a synonym for y, which also works
  • D - distance from origin length units
  • T - counter-clockwise angle (in radians) from the positive x axis (i.e., θ in polar coordinates)

Here (R,Z) are axisymmetric coordinates and (D,T) are polar coordinates, where D is distance from the origin to the (x,y) (or (R,Z) if axisymmetric) point (in length units) and T is counter-clockwise angle (in radians) from the positive x (or R if axisymmetric) axis to the point. These extra variables are only allowed in:

Function Format

Some details on entering functions are:

  • In scripted files, the function must be enclosed in quotes (e.g., "x^2+y^2") to prevent it from being evaluated as a command expression prior to being used in the analysis.
  • Enter variables simply as x, y, etc., and not with the preceding "#" used for command expression variables.
  • Operators: The function uses standard operators + - * / and ^ with standard operator precedence for addition, subtraction, multiplication, division, and raising to a power.
  • The function can contain the following defined functions:
    • sin(x), cos(x), and tan(x) - trigonometric function of angle in radians.
    • asin(x), acos(x), and atan(x) - inverse trigonometric function with result in radians.
    • sqrt(x) - square root.
    • log(x) and ln(x) - log base 10 and natural log, respectively (note these are different than log functions used in command expressions in scripted files due to different math expression parser in the code engines).
    • exp(x) - exponential of x.
    • abs(x) - absolute value of x.
    • int(x) - integer part of x as next lower integer. For negative numbers, it is next lower integer or int(-3.4)=-4.
    • Sinh(x), Cosh(x), and Tanh(x) - hyperbolic trigonometric functions. (note the initial uppercase is needed here, and differs from scripted command expressions due to different math expression parser in the code engines).
    • erf(x) and erfc(x) - error function and error function complement.
  • The following functions are helpful if creating boundary conditions:
    • [math]\displaystyle{ {\rm ramp}(A,x) = \left\{\begin{array}{ll} 0 & x\le 0 \\ Ax & 0\lt x\lt 1 \\ A & x \ge 1 \end{array}\right. }[/math]
    • [math]\displaystyle{ {\rm cosramp}(A,x) = \left\{\begin{array}{ll} 0 & x\le 0 \\ {A\over 2}\left(1-\cos(\pi x)\right) & 0\lt x\lt 1 \\ A & x \ge 1 \end{array}\right. }[/math]
    • [math]\displaystyle{ {\rm box}(A,x) = \left\{\begin{array}{ll} 0 & x\le 0 \\ A & 0\lt x\lt 1 \\ 0 & x \ge 1 \end{array}\right. }[/math]
    • [math]\displaystyle{ {\rm sinbox}(A,x) = \left\{\begin{array}{ll} 0 & x\le 0 \\ A\sin(\pi x) & 0\lt x\lt 1 \\ 0 & x \ge 1 \end{array}\right. }[/math]
    • [math]\displaystyle{ {\rm sgn}(x) = \left\{\begin{array}{ll} -1 & x \lt 0 \\ 0 & x = 0 \\ +1 & x \gt 0 \end{array}\right. }[/math]
    • [math]\displaystyle{ {\rm sign}(x) = \left\{\begin{array}{ll} 0 & x \lt 0 \\ 1 & x \ge0 \end{array}\right. }[/math]
    • [math]\displaystyle{ {\rm tri}(x) = \left\{\begin{array}{ll} 1-|x| & |x| \lt 1 \\ 0 & |x| \ge1 \end{array}\right. }[/math]
    • [math]\displaystyle{ {\rm mod}(x,y) = {\rm fmod}(x,y) = x-{\rm floor}(x/y)*y }[/math]

    For example, to apply a ramp from 0 to tf, use x = t/tf; to apply a ramp from ts to tf,use x = (t-ts)/(tf-ts).

  • Functions can include pi (or Pi or PI) for the number π.
  • Exponential Notation: numbers can have "e" or "E" for powers of ten such as 1.4e3 for 1400.
  • Extra spaces in the function are ignored.

Notes

Integrals and Derivatives

The cosramp(A,x) and sinbox(A,x) are useful for gradually applying velocity or displacement at the start of the simulation rather than instantaneously applying some velocity or displacement at time zero. To interchange between velocity and displacement, it is useful to evaluate their integrals and derivatives:

     [math]\displaystyle{ \int_0^x {\rm sinbox}(A,u)du = {A\over\pi}\bigl(1-\cos(\pi x)\bigr) = {\rm cosramp}\left({2A\over\pi},x\right) }[/math]

     [math]\displaystyle{ \int_0^x {\rm cosramp}(A,u)du = \left\{\begin{array}{ll} {A\over 2}\left(x-{\sin(\pi x)\over \pi}\right) & x\lt 1 \\ A\left(x-{1\over 2}\right) & x \ge 1 \end{array}\right. = {A\over 2}\bigl(x + (x-1){\rm sign}(x-1)\bigr) - {\rm sinbox}\left({A\over2\pi},x\right) }[/math]

     [math]\displaystyle{ {d\bigl({\rm cosramp}(A,x)\bigr)\over dx} = {A\pi\over 2}\sin(\pi x) = {\rm sinbox}\left({A\pi\over 2},x\right) }[/math]

     [math]\displaystyle{ {d\bigl({\rm sinbox}(A,x)\bigr)\over dx} = \left\{\begin{array}{ll} 0 & x\le0 \\ {A\pi}\cos(\pi x) & 0\lt x\lt 1 \\ 0 & x \ge 1 \end{array}\right. = \pi A\thinspace {\rm sign}(x) - {\rm cosramp}(2\pi A,x) }[/math]

Cosine Ramp to Desired Velocity

In problems with constant velocity loading, it might be preferable to ramp velocity to the desired velocity V over some ramp time tr rather start the problem at time zero with velocity V. The ramped-up velocity can be done using the cosramp(A,x) function by setting A = V and x = t/tr or:

     [math]\displaystyle{ v(t) = {\rm cosramp}\left(V,{t\over t_r}\right) }[/math]

Integrating this velocity, the resulting displacement is:

     [math]\displaystyle{ d(t) = \left\{\begin{array}{ll} {V\over 2}\left(t-{t_r\over \pi}\sin\left({\pi t\over t_r}\right)\right) & t\lt t_r \\ V\left(t-{t_r\over 2}\right) & t \ge t_r \end{array}\right. }[/math]

In terms of defined functions, this displacement can be written

    [math]\displaystyle{ d(t) = {V\over2}\left[t + (t-t_r){\rm sign}(t-t_r) - {\rm sinbox}\left({t_r\over\pi},{t\over t_r}\right)\right] }[/math]

For example, to ramp to V and then hold constant velocity until reaching desired displacement dmax, the simulation time should be set to:

    [math]\displaystyle{ t_{max} = {d_{max}\over V} + {t_r\over 2} }[/math]

The first term is the simulation time using constant velocity while the second term is the extra time needed because the velocity was ramped to velocity V over time tr.

Cosine Ramp to Desired Displacement

Rather that apply displacement instantaneously, which would be an impact problem, it might be preferable to ramp to displacement d over ramp time tr. A displacement function would be

     [math]\displaystyle{ d(t) = {\rm cosramp}\left(d,{t\over t_r}\right) }[/math]

Differentiating this displacement to get velocity that must be applied as a boundary condition gives

     [math]\displaystyle{ v(t) = {\pi d\over 2t_r}\sin\left({\pi t\over t_r}\right) = {\rm sinbox}\left({\pi d\over 2t_r},{t\over t_r}\right) }[/math]

The maximum velocity used during this loading is

     [math]\displaystyle{ v_{max} = {\pi d\over 2t_r} }[/math]

Linear Ramp to Desired Displacement

To ramp to a displacement d over time tr with a constant velocity, the ramp velocity is

     [math]\displaystyle{ v(t) = {\rm box}\left({d\over t_r},{t\over t_r}\right) }[/math]

The displacement corresponding to this velocity is

     [math]\displaystyle{ d(t) = {\rm ramp}\left(d,{t\over t_r}\right) }[/math]

Loading and Unloading Cycles

To ramp at velocity vr from t0 and t0+tr then unload using the reverse velocity from t0+tr to t0+2tr, the velocity is:

     [math]\displaystyle{ v(t) = {\rm box}\left(v_r,{t-t_0\over t_r}\right)-{\rm box}\left(v_r,{t-t_0-t_r\over t_r}\right) = {\rm box}\left(v_r,{t-t_0\over 2t_r}\right){\rm sgn}(t_0+t_r-t) }[/math]

The displacement corresponding to this velocity is

     [math]\displaystyle{ d(t) = v_rt_r {\rm tri}\left({t-t_0\over t_r}-1\right) }[/math]

A sequence of loading and unloading steps can be created by defining periodic function as explained in the next section.

Periodic Boundary Condition Function

The mod(A,x) function is useful for defining any periodic boundary condition function of time. First create any function of time that is well defined over a given cycle time tc. To define a periodic function replace every "t" in that function with "mod(t,tc)". The result will be a periodic function that will repeat the same defined cycle as long as the simulations keeps running. For example, a sequence of triangular displacements from the previous section's example staring at the beginning (t0=0) could be defined with:

     [math]\displaystyle{ v(t) = {\rm box}\left(v_r,{{\rm mod}(t,2t_r)\over t_r}\right)-{\rm box}\left(v_r,{{\rm mod}(t,2t_r)-t_r\over t_r}\right) }[/math]

Rigid Rotation

See Rigid Material documentation for details on moving a plot of rigid particles in rigid rotation about a point.

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