Physics2D / computed foundations

Same dots. Different material.

Shear the patch without changing its area, then compress it. The missing ingredient is the material law—not whether the representation contains dots.

All three patches receive exactly the same prescribed deformation. Arrows show stress traction on the patch boundary. A liquid at rest resists compression but has no static shear stiffness; an elastic solid also resists distortion. The dashed square is the original shape.
—Area ratio J
—Liquid pressure
—Jelly shape energy
Computed teaching model
Equations and model boundary
ρ/ρ₀ = 1/J;  pliquid = K max(1/J − 1, 0)
Wbulk = K(J − ln J − 1)
Wsolid = Wbulk + μ ‖F − R‖²

A constitutive-response comparison, not a time-stepping SPH/MPM benchmark. R is the polar rotation of F; rigid rotation adds no elastic energy. The displayed Wbulk is the solid bulk energy. For the liquid, it is clamped to zero at J≥1, matching the absence of tensile pressure. Shape energy includes elastic stretch as well as shear. Liquid viscosity and surface tension are omitted to isolate static response. The independent-point control has neither a density law nor stored strain; pairwise contacts alone do not supply either. SPH and MPM are numerical representations for continuum laws, not substitutes for choosing those laws.