# Finite Element Method

Finite-element surrogates are usually trained and compared by displacement error, although engineering decisions rest on stresses. This study replaces the norm of the supervised loss with the energy norm of the assembled stiffness matrix, leaving everything else fixed; in pre-registered comparisons a three-dimensional transformer's median von Mises error fell 6.1-fold and the two-dimensional mean fell by more than half, also on held-out instances, and training on the energy alone without solutions came close to that accuracy.

Finite-element surrogates are usually trained and compared by displacement error, although engineering decisions rest on stresses. This study replaces the norm of the supervised loss with the energy norm of the assembled stiffness matrix, leaving everything else fixed; in pre-registered comparisons a three-dimensional transformer's median von Mises error fell 6.1-fold and the two-dimensional mean fell by more than half, also on held-out instances, and training on the energy alone without solutions came close to that accuracy.

A finite-element solver for conjugate heat transfer and buoyant flow is recast as fixed-weight graph programs inside an automatic-differentiation framework, with assembled operators and the algebraic-multigrid cycle executed as sparse graph operations and nothing trained. With discretization, mesh, time step and tolerances held fixed, the same problem runs up to 181 times faster per step on one H200 than on 32 CPU cores, and the pressure solve alone 18.7 times faster. Reverse-mode gradients match central differences, and implicit differentiation holds one gradient near 4.9 forward solves.

A finite-element solver for conjugate heat transfer and buoyant flow is recast as fixed-weight graph programs inside an automatic-differentiation framework, with assembled operators and the algebraic-multigrid cycle executed as sparse graph operations and nothing trained. With discretization, mesh, time step and tolerances held fixed, the same problem runs up to 181 times faster per step on one H200 than on 32 CPU cores, and the pressure solve alone 18.7 times faster. Reverse-mode gradients match central differences, and implicit differentiation holds one gradient near 4.9 forward solves.

Under a uniform dilatational eigenstrain, an ellipsoidal inclusion has a spatially uniform internal strain; a two-sided inequality now links how far a shape sits from the ellipsoid family to how heterogeneous its interior strain becomes, with a half-order Sobolev scale that brings morphological wavelength into the mechanics. For protruded, indented, and necked geometries, ellipsoidal subtraction yields an exact residual-field budget whose bounds tighten monotonically and agree with independent displacement-based finite-element results.

Under a uniform dilatational eigenstrain, an ellipsoidal inclusion has a spatially uniform internal strain; a two-sided inequality now links how far a shape sits from the ellipsoid family to how heterogeneous its interior strain becomes, with a half-order Sobolev scale that brings morphological wavelength into the mechanics. For protruded, indented, and necked geometries, ellipsoidal subtraction yields an exact residual-field budget whose bounds tighten monotonically and agree with independent displacement-based finite-element results.