Symbolic Math (Shape Functions + Exact Stiffness Integral)

Showcase & Examples
1

Symbolic math is a powerful tool when you need exact expressions instead of approximate numerics. It’s especially valuable for:

  • high-precision modeling (no rounding until you decide)

  • automatic algebraic simplification

  • differentiation/integration without manual derivations

  • generating clean formulas for documentation (LaTeX) or code generation

This example demonstrates a typical engineering workflow: deriving cubic beam shape functions and computing an exact stiffness-related integral symbolically, then evaluating the shape functions numerically over a grid.

What this example does

  • Loads the symbolic engine (sym.load())

  • Defines symbolic variables (le, x, E, Iz)

  • Builds a polynomial basis and boundary-condition matrix P

  • Computes shape functions: N = p * inv(P)

  • Computes second derivatives: d2N = diff(N, x, 2)

  • Computes an internal stiffness term via an exact integral.

  • Substitutes le = 1 and evaluates N(x) over xi = 0..1

  • Displays symbolic results as LaTeX

JSLAB_symbolic
JSLAB_symbolic1139×707 68.7 KB

JavaScript Example

// Symbolic Math

var le, x, E, Iz;
var p, P, invP, N, d2N;
var k_int;
var xi = range(0, 1, 0.01);

await sym.load();
[le, x, E, Iz] = sym.syms(['le', 'x', 'E', 'Iz']);

P = sym.mat([
  [1, 0, 0, 0],
  [0, 1, 0, 0],
  [1, le, sym.pow(le, 2), sym.pow(le, 3)],
  [0, 1, sym.mul(2, le), sym.mul(3, sym.pow(le, 2))]
]);

p = sym.mat([[1, x, sym.pow(x, 2), sym.pow(x, 3)]]);

invP = sym.inv(P);
N = sym.mul(p, invP);
d2N = sym.diff(N, 'x', 2);

k_int = sym.mul(
  E, Iz,
  sym.intg(sym.mul(sym.transp(d2N), d2N), x, [0, le])
);

// Evaluate N(x) numerically for le=1 on xi grid
Ni = sym.subs(sym.subs(N, le, 1), x, xi).toNumeric();
var N_flat = Ni.flat();

// Show results as LaTeX
sym.showLatex(N);
sym.showLatex(k_int);

Practical uses

  • Finite element derivations: shape functions, stiffness/mass matrices, exact integrals

  • Control and estimation: symbolic Jacobians/Hessians for nonlinear models

  • Education & documentation: show derivations as LaTeX directly from code

  • Verification: compare numeric implementations against exact symbolic ground truth