Source-linked AI summary
A NURBS-based Inverse Analysis for Reconstruction of Nonlinear Deformations of Thin Shell Structures
N. Vu-Bac, T. X. Duong, T. Lahmer, X. Zhuang, R. A. Sauer, H. S. Parke, T. Rabczuk
TL;DR
The paper addresses inverse reconstruction of thin-shell loads and deformations when geometric and material nonlinearities, including instabilities, are present. It combines NURBS-based shell discretization with gradient-based optimization using displacement data at discrete locations. Numerical examples show accurate recovery of target shapes and unstable buckling-related deformations, while uniqueness is only established numerically for simplified problems.
Problem
Existing inverse studies rarely reconstruct three-dimensional bending deformations with high-fidelity plate or shell models, especially when nonlinear instabilities are allowed.
Method
The method combines NURBS-based finite elements, nonlinear kinematics and constitutive models, and gradient-based optimization with analytical or semi-analytical sensitivities.
Results
The proposed inverse method accurately identifies applied loads and reconstructs stable or unstable shell deformations from experiment-like displacement data.
Takeaways & Limitations
The approach captures snap-through, snap-back, and buckling in numerical shell reconstructions and is presented as applicable to target-shape recovery.
Takeaways & Limitations
Uniqueness is supported numerically only for simplified problems; generalization to problems with more degrees of freedom is not established.
Abstract
from arXiv · showhide
This article presents original work combining a NURBS-based inverse analysis with both kinematic and constitutive nonlinearities to recover the applied loads and deformations of thin shell structures. The inverse formulation is tackled by gradient-based optimization algorithms based on computed and measured displacements at a number of discrete locations. The proposed method allows accurately recovering the target shape of shell structures such that instabilities due to snapping and buckling are captured. The results obtained show good performance and applicability of the proposed algorithms to computer-aided manufacturing of shell structures.
1. Introduction
The paper addresses inverse reconstruction of thin-shell loads and deformations, including unstable shape changes, within a gap in high-fidelity three-dimensional shell inverse analysis. It proposes a NURBS-based, gradient-optimized approach combining geometric and material nonlinearities.
- Motivation: Thin structures can undergo complex three-dimensional shape changes under growth, swelling, heating, or external loads.Understanding and controlling these changes is relevant to biomimetic engineering.
- Motivation: Inverse analysis determines the stimuli required to produce a target shape and reconstructs the resulting displacement field.The approach is used for target-shape reconstruction of thin sheets and shells.
- Research gap: Few studies reconstruct three-dimensional bending deformations, and even fewer address high-fidelity plate or shell structures.Earlier work included least-squares reconstruction from strain measurements, but the literature remains limited for detailed shell models.
- Contribution: NURBS discretization supplies the C1-continuity required by Kirchhoff-Love shells and avoids rotational degrees of freedom.The paper selects NURBS for lower computational cost and better modeling of complex geometries.
- Contribution: The proposed inverse approach accounts for both geometric and material nonlinearities while allowing snap-through, snap-back, and buckling.It determines external loads and reconstructs deformations from given data.
- Approach: Gradient-based optimization with analytical and semi-analytical sensitivities forms the basis of the inverse analysis.The paper presents numerical examples to verify the method and examine elastic instabilities.
2. A brief description of rotational-free thin shell theory
The shell kinematics maps points from a parameter domain through reference and current surfaces, then derives tangent, normal, metric, curvature, and deformation measures from those mappings.
- Surface mapping: The reference and current shell surfaces are represented parametrically over a common parameter domain.The reference surface S0 and current surface S describe the shell before and after deformation.
- Surface mapping: Covariant tangent vectors are obtained by differentiating the surface mappings with respect to the parameter coordinates.These vectors define the local surface geometry.
- Surface geometry: Surface normals are constructed from cross products of the two covariant tangent vectors in the reference and current configurations.The resulting normalized vectors distinguish the undeformed and deformed surface orientations.
- Surface geometry: Metric tensors are formed from tangent-vector inner products, while curvature tensors characterize the surface bending geometry.Mean, Gaussian, and principal curvatures are then obtained from the curvature tensor.
- Deformation measures: The surface deformation gradient maps the reference surface to the current surface and generates the right and left Cauchy-Green tensors.These measures encode surface deformation relative to the reference configuration.
2.2. Balance laws
The balance-law formulation expresses quasi-static shell equilibrium through surface forces, moments, boundary conditions, and a weak form that is linearized for numerical solution.
- Equilibrium: Quasi-static equilibrium is formulated for a shell surface subjected to prescribed body forces.The formulation introduces surface tractions and stress-resultant quantities.
- Stress resultants: Membrane stresses and bending moments define the in-plane and bending components of the shell’s distributed sectional forces.These quantities enter the shell balance laws and boundary conditions.
- Boundary conditions: Boundary conditions prescribe displacements, rotations, tractions, and bending moments on corresponding boundary segments.The bending moments include components parallel and perpendicular to the boundary.
- Weak form: The weak form seeks the shell configuration whose internal and external virtual work satisfy equilibrium for all admissible variations.Point-load virtual work can contribute at boundary corners.
- Numerical solution: Linearization produces incremental internal and external virtual-work equations that support Newton-Raphson and arc-length solution methods.These methods solve the resulting linearized system of equations.
2.4. Constitutive equations
The constitutive formulation uses hyperelastic shell models derived from strain-energy functions, including Neo-Hookean and Koiter descriptions of stretching and bending.
- Hyperelasticity: Hyperelastic constitutive equations for stretching and bending are derived from a strain-energy function.The Neo-Hookean model describes nonlinear stress-strain behavior under large deformations and is non-dissipative.
- Koiter model: The Koiter model defines surface strain energy for initially curved shells.Its formulation represents shell constitutive behavior through strain and curvature measures.
- Koiter model: The Koiter constitutive tensors combine material parameters with membrane and bending contributions.The formulation introduces tensors C and F through material constants Λ and µ.
- Material reduction: The material parameters Λ and µ can be obtained by analytically integrating a three-dimensional Saint Venant-Kirchhoff model through the shell thickness.The resulting parameters relate to the classical three-dimensional Lamé constants.
2.6. Shell constitutive model derived from 3D material model
The shell constitutive model projects a three-dimensional continuum formulation onto the shell midsurface. Through-thickness integration then provides projected Kirchhoff stresses and bending moments for the two-dimensional shell model.
- Shell constitutive model: The 3D continuum model is projected onto the reference and deformed midsurfaces to obtain a 2D Kirchhoff-Love shell model.The mapping uses the actual metric at an arbitrary material point through the shell thickness.
- Through-thickness projection: The thickness coordinate spans ξ ∈ [−T/2, T/2], with d := λ3n and λ3 denoting the normal-direction stretch.This thickness description is used in the Kirchhoff-Love shell model.
- Through-thickness projection: Projected Kirchhoff stresses and bending moments are obtained from the three-dimensional Kirchhoff stress components through the detailed projection procedure.The projected quantities enter the shell constitutive description after integrating or projecting the 3D response across thickness.
2.7. Compressible Neo-Hookean materials
The material formulation introduces a compressible three-dimensional Neo-Hookean model and evaluates shell stresses and moments through projected metric quantities. The article considers both Koiter and projected constitutive models.
- Compressible Neo-Hookean material: The compressible Neo-Hookean formulation defines the three-dimensional strain energy per reference configuration.Its invariants are those of the three-dimensional Cauchy-Green tensor.
- Compressible Neo-Hookean material: The three-dimensional Kirchhoff stress is derived from the Neo-Hookean strain-energy formulation.The stress expression is subsequently used to evaluate the shell stress and moment quantities.
- Compressible Neo-Hookean material: The covariant and contravariant metric tensors at a material point are defined from covariant and contravariant tangent vectors.These metric quantities support the constitutive projection from the three-dimensional material response.
- Constitutive models: The article considers two constitutive material models: the Koiter model and the projected model based on the shell projection equations.The projected model uses the stress and moment relations derived from the three-dimensional formulation.
- Surface discretization: NURBS shape functions discretize the shell surface because Kirchhoff-Love formulations require C1-continuity.The NURBS discretization is used to solve the shell weak-form equation and interpolate geometry through control points.
2.9. Discretized weak form
The weak form is discretized using the shell interpolations, producing element-level internal and external virtual-work contributions. These contributions account for membrane, bending, body-force, pressure, traction, and boundary-moment effects.
- Weak-form discretization: The weak form is discretized using the preceding shell interpolations to obtain a finite-element approximation.The resulting assembly is expressed over the number of elements.
- Internal virtual work: Element internal virtual work is represented by finite-element force vectors associated with membrane stress and bending moment.The membrane and bending contributions are defined elementwise.
- External virtual work: Element external virtual work includes constant body force, pressure normal to the surface, boundary traction, and boundary moment.These loads define the external finite-element force vectors.
2.10. Lagrange multiplier method for rotational constraints
Rotational continuity between shell patches is enforced by adding a constraint potential to the shell formulation. The constraint uses a Lagrange multiplier and compares neighboring-patch normals in reference and deformed configurations.
- Rotational constraint: The method enforces G1-continuity between patches by adding a rotational constraint potential to the shell formulation.The constraint is introduced specifically to control patch-boundary rotations.
- Rotational constraint: The rotational constraint uses p as a Lagrange multiplier.The multiplier enters through the added constraint potential.
- Normal alignment: The constraint compares angles between neighboring-patch normals in the reference and deformed configurations.The reference normals are N and ¯N, while n and ¯n denote the corresponding deformed normals; L0 identifies the relevant reference patch boundary.
3. Inverse analysis
The inverse procedure identifies shell-loading and boundary-condition variables from measured surface displacements while accounting for nonlinear equilibrium and instability-driven shape changes. It combines NURBS-based thin-shell analysis with gradient-based sensitivities, including analytical, adjoint, and finite-difference routes.
- Inverse formulation: The procedure identifies applied loads and prescribed boundary conditions that produce a desired thin-shell shape from measured displacements.Design variables can represent external loads, prescribed displacements, tractions, moments, or lateral pressure.
- Inverse formulation: The objective is a least-squares mismatch between computed and measured surface displacements, subject to inequality and side constraints.The state variables are nodal displacements, while the design variables are bounded by lower and upper limits.
- Scope: The formulation permits reconstruction of nonlinear shell deformations through instability-related shape changes such as snapping and buckling.The method is presented for an isogeometric, rotation-free thin shell with hyperelastic material behavior and nonlinear kinematics.
- Sensitivity analysis: Nonlinear equilibrium is differentiated with respect to design variables, and gradient-based optimization uses adjoint vectors to avoid explicitly computing displacement derivatives.The tangent stiffness matrix and derivatives of internal and external forces enter the sensitivity calculation.
- Sensitivity analysis: Analytical sensitivities are derived for prescribed displacements, edge tractions, point loads, distributed moments, and lateral pressure.For prescribed displacements, the relevant tangent-stiffness columns correspond to the design-vector degrees of freedom; point-load sensitivities use the nonzero loaded-node entry.
- Sensitivity analysis: Semi-analytical sensitivities use finite differences with a perturbation step of 0.001 × s_i to limit truncation error.The perturbed quantities are evaluated in the direction of each design variable.
4. Numerical examples
Numerical examples show that the inverse formulation recovers applied loads, prescribed displacements, and nonlinear shell shapes from experiment-like displacement data. The method reconstructs configurations involving snap-through and buckling, while noise and inverse-problem uniqueness constrain its scope.
- Validation procedure: The validation uses computed shell displacements at discrete locations, perturbed by 1% random noise, as experiment-like measurements for gradient-based inverse optimization.The study uses a two-step procedure because experimental measurements are unavailable.
- Benchmark problems: Four numerical examples cover cantilevered and hinged cylindrical shells with geometrically nonlinear mechanics, hyperelastic constitutive laws, and instability cases involving snap-through and buckling.The examples include prescribed end displacement or shear traction, prescribed center displacement, and imperfect shells under compression.
- Cantilever shell: The cantilever inverse solutions recover t_inverse = 1.4927 N/mm and w_inverse_tip = 4.0077, closely matching t_meas = 1.5 N/mm and w_meas_tip = 4.The reconstructed displacement-controlled configuration has w_inverse_tip = 3.9696 and a maximal error of 0.76% at free-end nodes.
- Cantilever shell: Increasing systematic noise γ decreases algorithm accuracy, while the objective function is convex with respect to the cantilever end displacement w_tip for the simplified problem.The numerical uniqueness assessment relies on identifying a clear minimum; generalization to problems with more degrees of freedom is not established.
- Hinged cylindrical shell: For the hinged cylindrical shell, the reconstructed center displacement is w_inverse_cen = −24.90 after 20 iterations, with good agreement achieved despite snap-through instability.The forward Koiter response also agrees with the reported Saint Venant-Kirchhoff reference response.
- Instability reconstruction: The later instability examples recover prescribed displacements, moments, lateral pressure, and buckling-induced target shapes nearly exactly under both Koiter and projected material models.Reported reconstructions include v_inverse_proj = 11.13 × 10^3 N.mm/mm and v_inverse_proj = 35.74 N/mm^2, with the specified buckling shapes recovered.
5. Conclusions
The developed inverse method combines NURBS-based finite elements with material and geometric nonlinearities to identify applied loads and reconstruct thin-shell deformations, including stable and unstable shape changes.
- NURBS-based finite elements provide geometrically exact discretization and C1-continuity for the three-dimensional shell surface.
- The method combines Koiter and projected compressible Neo-Hookean shell models with gradient-based optimization using analytical and semi-analytical sensitivities.
- Numerical simulations recover thin-shell target shapes under stable deformation, snap-through, snap-back, and buckling.
- The inverse analysis identifies unknown applied loads from displacement data and reconstructs deformations with high accuracy.
- The authors position the approach as a potential basis for computer-aided manufacturing of shell structures.
Appendix A. FE tangent matrices
The appendix presents tangent-matrix components and figure-based evaluations for flat strips under compression, bending moments, and lateral disturbing pressure using Koiter and projected models.
- Figures 15 and 17 track objective-function convergence and reconstructed flat-strip shapes across iterations for Koiter and projected models.
- The tangent-matrix formulation includes material stiffness matrices, geometric stiffness matrices, external moment, and surface-pressure components.
- Figures 18–22 examine axial force-displacement responses, deformed shapes, reaction forces, and inverse reconstruction under lateral disturbing pressure.
- The curve-boundary coordinate ξ is used in defining the relevant element-level tangent-matrix expressions.