Interactive Elastic Sheet

The homepage grid is a reduced linear-elastic calculation, not a geometric warp. The reference sheet is a rectangle Omega = [0, Lx] x [0, Ly] whose outer boundary is fixed.

Kinematics and energy

The model uses small-strain plane-stress elasticity with mu = 1, nu = 0.35, and

lambda = 2 mu nu / (1 - nu).

For a displacement field u, the strain is evaluated analytically as

epsilon = 1/2 (grad u + grad u^T),
W = mu epsilon:epsilon + lambda/2 (tr epsilon)^2.

Both displacement components are expanded in the fixed-boundary sine basis

phi_mn(x, y) = sin(m pi x / Lx) sin(n pi y / Ly),
u_x = sum q_mn^(x) phi_mn,   u_y = sum q_mn^(y) phi_mn.

Every basis function vanishes on the four outer edges, so u = 0 on the entire boundary exactly. The homepage uses an 8 x 8 scalar basis, or 128 displacement degrees of freedom.

Local drag constraint

Mouse-down defines a smooth, normalized, truncated Gaussian handle w_a centered at the grab location. A drag vector D constrains the handle-average displacement,

integral_Omega w_a(x) u(x) dA = D.

The reduced stiffness is assembled directly from

K_IJ = integral_Omega [2 mu epsilon(Phi_I):epsilon(Phi_J)
       + lambda div(Phi_I) div(Phi_J)] dA.

For each new anchor, the code builds the two-row constraint matrix B using 48-point Gauss-Legendre integration of the normalized Gaussian. The field is the unique discrete minimum of elastic energy subject to the constraint:

q = K^-1 B^T (B K^-1 B^T)^-1 D.

K is assembled and Cholesky-factorized only when the canvas size changes. On pointer-down, two back-solves construct the two unit response fields. While dragging, the displayed result is only their linear combination; no new matrix system is solved per animation frame.

Rendering and validity

The grid is drawn from the resulting spectral displacement field. Strain and principal strains are evaluated from the analytic derivatives of the same basis, not from screen-space line spacing. A low-opacity tensile/compressive tint is rendered beneath the grid.

The drag is capped by the largest principal strain sampled from a 33 x 25 analytic response field. The default cap is 7.5%, keeping the display inside a visually useful small-strain regime. The return-to-rest motion is a UI interpolation only; every intermediate displayed state is still the constrained static linear-elastic solution for its interpolated handle displacement.

Numerical checks

ElasticSheet.runValidation() tests the following for representative central, off-center, and near-boundary handles:

1. Boundary displacement is zero to floating-point precision.
2. The two averaged handle constraints satisfy B q = D to floating-point precision.
3. The stiffness matrix is exactly symmetric in the assembled representation and accepts a positive-pivot Cholesky factorization.
4. Strain is computed as sym grad u, so compatibility is built into the spectral displacement representation.
5. Feasible null-constraint perturbations increase the discrete elastic energy and are K-orthogonal to the minimizer.
6. 5 x 5, 6 x 6, and 8 x 8 bases are compared on common sample points against a 10 x 10 reference.
7. The capped response is sampled for det(I + grad u) to catch folds or pathological near-boundary behavior.

Validation results

The following values were obtained by running ElasticSheet.runValidation() for four representative handles: central, two off-center, and one near a fixed edge. Drag directions were capped at the same 7.5% principal-strain limit used by the interface.

The relative displacement differences to the 10 x 10 reference were:

The default resolution was therefore kept at 8 x 8: it is visually smooth near the local handle and close to the higher-order reference without making animation work noticeable. A local timing check on a normal desktop runtime gave about 4.5 ms to construct and factorize the 128 x 128 system (only after resize), 0.15 ms to prepare a new handle, and about 0.23 ms for the displacement evaluations used by one grid redraw. Drag frames do not factorize or solve K; they only combine the two cached unit response fields and evaluate the spectral series.

Repeating the same checks for the square-like mobile canvas gave boundary and constraint residuals below 1.3e-17, a minimum sampled det(I + grad u) of 0.931, and an 8 x 8 mean/reference difference of 1.18% (maximum 2.65%).