arXiv subjects

Cristian G. Gebhardt

Publications and source records attributed to Cristian G. Gebhardt.

11 recordsLinked to original sources

arXiv2026

A data-driven solving strategy based on a greedy optimization algorithm for the analysis of nonlinear beam structures

In the last decade, data-driven computational mechanics (DDCM) has emerged as a novel paradigm in computational mechanics, enabling the direct use of constitutive data - such as stress-strain pairs obtained from experiments, without relying on ad-hoc material models and thereby avoiding information loss. In this work, we extend our data-driven solving strategy GO-ADM, which combines a greedy optimization algorithm with the alternating direction method (ADM), to the structural analysis of geometrically exact beams formulated using director-based kinematics. We discuss a data initialization strategy for nonlinear systems based on a conventional finite element analysis of the same structure using a prescribed constitutive model. The resulting discrete stress and strain fields, possibly obtained under multiple loading scenarios, may also be employed as artificial datasets for the subsequent data-driven computations. Furthermore, we investigate the thermomechanical consistency of both the dataset and the discrete solution, and propose a weak enforcement of this consistency in the latter via a penalty approach. Numerical examples involving single- and multi-member structures demonstrate that the proposed penalty term leads to thermomechanically consistent discrete stress and strain fields. Moreover, for the studied examples, the solving strategy GO-ADM yields a generally improved approximation of the globally optimal solution compared to the standard ADM-based direct solver.

cs.CE↗

arXiv2026

Intrinsic Structure of Elasticity in Hilbert Space

Mathematical elasticity has a long history and a huge body of literature. Surprisingly, at its heart elasticity exhibits a rich and transparent structure that, to our knowledge, is rarely presented in an explicit and unified form. Using standard tools from functional analysis, this article reveals four orthogonal decompositions of Hilbert spaces that characterize the static equilibrium of an elastic body at small deformations, universally for arbitrary spatial dimension, arbitrary boundary conditions, and classical as well as data-driven formulations. Regarding data-driven continuum mechanics, it highlights the intrinsic structure that remains unchanged when constitutive laws are replaced by material data sets.

math.AP↗

arXiv2026

A Data-Driven Computational Framework for Incompressible Flow in Hydraulic Networks

The classical procedure for solving hydraulic networks relies on the assumption of constitutive equations, which state the relationship between the pressure gradient and fluxes along an edge. In this paper, we propose a data-driven framework that bypasses these constitutive models, formulating the incompressible flow problem directly on the graph topology. By assigning discrete measured data points to network edges, the problem is cast as a mixed-integer quadratic optimization over nodal pressures, edgewise states, and data assignments, accommodating both laminar (convex) and turbulent (non-convex) regimes. To solve this, we evaluate three algorithms: a GPU-accelerated Brute Force method, the Alternating Direction Method (ADM), and Deterministic Annealing (DA). The Brute Force method certifies global optima for small networks, establishing a good baseline for the iterative solvers. We demonstrate that ADM is highly sensitive to its initialization, requiring a faithful surrogate model to avoid local minima. In contrast, DA eliminates this dependence through unsupervised clustering. By annealing the data assignment from the centroid to strict nearest-neighbor projections, DA consistently reaches the global optimum without prior manifold reconstruction. Furthermore, numerical experiments reveal that DA is robust to noisy data, remains thermodynamically admissible on all but the coarsest and noisiest datasets, and sustains its convergence rate on larger networks where Brute Force is intractable and ADM degrades. Finally, the framework is successfully validated on complex configurations, including mixed-component networks and a $958$-edge arteriovenous bed featuring a non-Newtonian Carreau--Yasuda model, demonstrating its scalability and practical applicability.

math.NA↗

arXiv2026

Solving strategies for data-driven one-dimensional elasticity exhibiting nonlinear strains

In this work, we extend and generalize our solving strategy, first introduced in [1], based on a greedy optimization algorithm and the alternating direction method (ADM) for nonlinear systems computed with multiple load steps. In particular, we combine the greedy optimization algorithm with the direct data-driven solver based on ADM which is firstly introduced in [2] and combined with the Newton-Raphson method for nonlinear elasticity in [3]. We numerically illustrate via one- and two-dimensional bar and truss structures exhibiting nonlinear strain measures and different constitutive datasets that our solving strategy generally achieves a better approximation of the globally optimal solution. This, however, comes at the expense of higher computational cost which is scaled by the number of "greedy" searches. Using this solving strategy, we reproduce the first cycle of the cyclic testing for a nylon rope that was performed at industrial testing facilities for mooring lines manufacturers. We also numerically illustrate for a truss structure that our solving strategy generally improves the accuracy and robustness in cases of an unsymmetrical data distribution and noisy data.

cs.CE↗

arXiv2026

Data-driven stress problem under purely normal homogeneous Neumann boundary conditions

Data-Driven Continuum Mechanics -- the continuous counterpart of Data-Driven Computational Mechanics -- is a modern paradigm that enhances classical continuum mechanics by incorporating finite sets of experimental material data directly, avoiding any form of constitutive modeling. Despite recent progress, its analytical foundations remain at an early stage. In this work, we establish a rigorous functional-analytic framework for the data-driven stress problem under purely homogeneous normal Neumann boundary conditions. The problem is formulated as finding a stress field (satisfying the balance of linear and angular momenta and the boundary conditions) that is closest, in an $L^p$-sense, to an auxiliary stress field that is simultaneously sought and locally resembles a finite discrete set of experimental stress states. Our analysis relies on two key ingredients. First, the divergence operator induces a topological isomorphism between the space of symmetric stress fields modulo its kernel and the space of loads balanced by rigid-body motions, ensuring the existence of an equilibrated response. Second, the finiteness of the material data set guarantees proximinality in the stress space, which in turn yields a complete existence and uniqueness theory for solution equivalence classes. Together, these two properties provide a rigorous mathematical foundation for the data-driven stress problem under purely homogeneous normal Neumann boundary conditions.

math.AP↗

arXiv2025

Numerical approximation of a PDE-constrained Optimization problem that appears in Data-Driven Computational Mechanics

We investigate an optimization problem that arises when working within the paradigm of Data-Driven Computational Mechanics. In the context of the diffusion-reaction problem, such an optimization problem seeks for the continuous primal fields (gradient and flux) that are closest to some predefined discrete fields taken from a material data set. The optimization is performed over primal fields that satisfy the physical conservation law and the geometrical compatibility. We consider a reaction term in the conservation law, which has the effect of coupling all the optimality conditions. We first establish the well-posedness in the continuous setting. Then, we propose stable finite element discretizations that consistently approximate the continuous formulation, preserving its saddle-point structure and allowing for equal-order interpolation of all fields. Finally, we demonstrate the effectiveness of the proposed methods through a set of numerical examples.

math.NA↗

arXiv2025

On the use of an advanced Kirchhoff rod model to study mooring lines

In this work, we investigate the application of an advanced nonlinear torsion- and shear-free Kirchhoff rod model, enhanced with a penalty-based barrier function (to simulate the seabed contact), intended for studying the static and dynamic behavior of mooring lines. The formulation incorporates conservative and non-conservative external loads, including those coming from the surrounding flow (added mass, tangential drag, and normal drag). To illustrate the favorable features of this model, we consider some key scenarios such as static configurations, pulsating force applications at the fairlead, and fluid-structure interaction between mooring lines and the surrounding flow. Verification against well-established solutions, including catenary configurations and OpenFAST simulations, shows excellent accuracy in predicting mooring line responses for a floating offshore wind turbine. Among the most important results, we can mention that under normal pulsating loads at the fairlead, the mooring line exhibits a transition from a drag-dominated regime at low frequencies to an added-mass-dominated regime at higher frequencies. Furthermore, tangential forcing at the fairlead reveals a strong coupling between axial and bending dynamics, contrasting with normal forcing scenarios where axial dynamics remain largely unaffected. These findings underscore the potential of the proposed approach for advanced mooring line simulations.

physics.flu-dyn↗

arXiv2024

Nonlinear dynamic analysis of shear- and torsion-free rods using isogeometric discretization and outlier removal

In this paper, we present a discrete formulation of nonlinear shear- and torsion-free rods introduced by Gebhardt and Romero in [20] that uses isogeometric discretization and robust time integration. Omitting the director as an independent variable field, we reduce the number of degrees of freedom and obtain discrete solutions in multiple copies of the Euclidean space (R^3), which is larger than the corresponding multiple copies of the manifold (R^3 x S^2) obtained with standard Hermite finite elements. For implicit time integration, we choose the same integration scheme as Gebhardt and Romero in [20] that is a hybrid form of the midpoint and the trapezoidal rules. In addition, we apply a recently introduced approach for outlier removal by Hiemstra et al. [26] that reduces high-frequency content in the response without affecting the accuracy, ensuring robustness of our nonlinear discrete formulation. We illustrate the efficiency of our nonlinear discrete formulation for static and transient rods under different loading conditions, demonstrating good accuracy in space, time and the frequency domain. Our numerical example coincides with a relevant application case, the simulation of mooring lines.

cs.CE↗

arXiv2021

A finite element method for simulating soft active non-shearable rods immersed in generalized Newtonian fluids

We propose a finite element method for simulating one-dimensional solid models moving and experiencing large deformations while immersed in generalized Newtonian fluids. The method is oriented towards applications involving microscopic devices or organisms in the soft-bio-matter realm. By considering that the strain energy of the solid may explicitly depend on time, we incorporate a mechanism for active response. The solids are modeled as Cosserat rods, a detailed formulation being provided for the special case of a planar non-shearable rod. The discretization adopts one-dimensional Hermite elements for the rod and low-order Lagrange two-dimensional elements for the fluid's velocity and pressure. The fluid mesh is boundary-fitted, with remeshing at each time step. Several time marching schemes are studied, of which a semi-implicit scheme emerges as most effective. The method is demonstrated in very challenging examples: the roll-up of a rod to circular shape and later sudden release, the interaction of a soft rod with a fluid jet and the active self-locomotion of a sperm-like rod. The article includes a detailed description of a code that implements the method in the Firedrake library.

math.NA↗

arXiv2019

A new conservative/dissipative time integration scheme for nonlinear mechanical systems

We present a conservative/dissipative time integration scheme for nonlinear mechanical systems. Starting from a weak form, we derive algorithmic forces and velocities that guarantee the desired conservation/dissipation properties. Our approach relies on a collection of linearly constrained quadratic programs defining high order correction terms that modify, in the minimum possible way, the classical midpoint rule so as to guarantee the strict energy conservation/dissipation properties. The solution of these programs provides explicit formulas for the algorithmic forces and velocities which can be easily incorporated into existing implementations. Similarities and differences between our approach and well-established methods are discussed as well. The approach, suitable for reduced-order models, finite element models, or multibody systems, is tested and its capabilities are illustrated by means of several examples.

math.NA↗

arXiv2019

Variational principles for nonlinear Kirchhoff rods

The present article studies variational principles for the formulation of static and dynamic problems involving Kirchhoff rods in a fully nonlinear setting. These results, some of them new, others scattered in the literature, are presented in a systematic way, helping to clarify certain aspects that have remained obscure. In particular, the study of transversely isotropic models reveals the delicate role that differential geometry plays in their formulation and unveils consequently some approximations that can be made to obtain simplified formulations.

math-ph↗