import { MatN, VecN } from '@holotope/core';
import { type SimplexConstitutiveEvaluationN } from './simplex-constitutive.js';
import { type SimplexConstitutiveHessianVectorEvaluationN } from './simplex-constitutive-curvature.js';
/** Isotropic St. Venant–Kirchhoff parameters in intrinsic material coordinates. */
export interface SimplexStVenantKirchhoffMaterialN {
    readonly firstLameParameter: number;
    readonly shearModulus: number;
}
/** Energy, stress, and current-position gradient for one simplex element. */
export interface SimplexStVenantKirchhoffEvaluationN extends SimplexConstitutiveEvaluationN<SimplexStVenantKirchhoffMaterialN> {
    /** `lambda tr(E) I + 2 mu E` in P11's rest-material basis. */
    readonly secondPiolaStress: MatN;
}
/**
 * Evaluates an isotropic St. Venant–Kirchhoff material on a k-simplex in RN.
 *
 * This is a constitutive CPU reference, not a solver or inversion barrier.
 * Full-dimensional orientation evidence remains available on `deformation`.
 */
export declare function evaluateSimplexStVenantKirchhoffN(restPositions: readonly VecN[], currentPositions: readonly VecN[], material: SimplexStVenantKirchhoffMaterialN): SimplexStVenantKirchhoffEvaluationN;
/**
 * Evaluates the exact matrix-free StVK potential Hessian along one direction.
 *
 * The returned products are derivatives of the vertex gradients, with the
 * mathematical sign `Hessian(U) * direction`.
 *
 * This is the optional capability that lets a compiled objective containing
 * StVK elements answer
 * `evaluateXpbdIncrementalPotentialAnalyticHessianVectorN` exactly. A law
 * without it still works as a force provider, but is named by that routine's
 * completeness preflight rather than silently contributing nothing.
 *
 * @example
 * A unit tetrahedron stretched along x, with one vertex pulled further the
 * same way. The products are the curvature felt at each vertex, and they
 * sum to zero: the element's internal response has nowhere else to go:
 * ```ts
 * const rest = [
 *   new VecN([0, 0, 0]), new VecN([1, 0, 0]),
 *   new VecN([0, 1, 0]), new VecN([0, 0, 1])
 * ];
 * const current = [
 *   new VecN([0, 0, 0]), new VecN([1.2, 0, 0]),
 *   new VecN([0, 1, 0]), new VecN([0, 0, 1])
 * ];
 * const directions = [
 *   new VecN([0, 0, 0]), new VecN([1, 0, 0]),
 *   new VecN([0, 0, 0]), new VecN([0, 0, 0])
 * ];
 *
 * const curvature = evaluateSimplexStVenantKirchhoffHessianVectorN(
 *   rest, current, directions, { firstLameParameter: 1, shearModulus: 1 }
 * );
 *
 * curvature.products.map((p) => p.data[0]); // [-0.83, 0.83, 0, 0]
 * curvature.netProductResidual; // 0
 * ```
 *
 * @example
 * Translation invariance is exact rather than approximate. Moving every
 * vertex the same way cannot change the stored energy, so the curvature
 * along a rigid translation vanishes identically — a cheap check that a
 * material's analytic derivative is the derivative of its own energy:
 * ```ts
 * const rest = [
 *   new VecN([0, 0, 0]), new VecN([1, 0, 0]),
 *   new VecN([0, 1, 0]), new VecN([0, 0, 1])
 * ];
 * const current = [
 *   new VecN([0, 0, 0]), new VecN([1.2, 0, 0]),
 *   new VecN([0, 1, 0]), new VecN([0, 0, 1])
 * ];
 * const translation = Array.from({ length: 4 }, () => new VecN([1, 0, 0]));
 *
 * const curvature = evaluateSimplexStVenantKirchhoffHessianVectorN(
 *   rest, current, translation, { firstLameParameter: 1, shearModulus: 1 }
 * );
 *
 * curvature.products.every((p) => p.lengthSq() === 0); // true, exactly
 * ```
 */
export declare function evaluateSimplexStVenantKirchhoffHessianVectorN(restPositions: readonly VecN[], currentPositions: readonly VecN[], directions: readonly VecN[], material: SimplexStVenantKirchhoffMaterialN): SimplexConstitutiveHessianVectorEvaluationN<SimplexStVenantKirchhoffEvaluationN>;
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