Source-linked AI summary
Data-driven computational mechanics
Trenton Kirchdoerfer, Michael Ortiz
TL;DR
The paper addresses uncertainty introduced by empirical material modeling in computational mechanics. It proposes data-driven computing, which directly combines experimental material data with conservation-law constraints, and evaluates its convergence and robustness. The resulting solutions converge to classical solutions as the data set approaches a classical material law, while finite-element solutions converge jointly with mesh size and data-set approximation.
Problem
Empirical material modeling can introduce error and uncertainty, while prior data-integrating approaches typically augment or identify models rather than replace them.
Method
Data-driven computing minimizes phase-space distance to experimental material data subject to compatibility, equilibrium, and other conservation-law constraints.
Results
Data-driven solutions converge to classical solutions as the data set increasingly approximates a classical material law, with joint convergence for finite-element linear elasticity under mesh and data-set refinement.
Takeaways & Limitations
The variational data-driven formulation generalizes classical problems to material behavior represented by arbitrary phase-space data sets, recovering classical solutions when the data set is a material-law graph.
Takeaways & Limitations
The paradigm is formulated for computational mechanics and elastic quasistatic problems; inelastic materials require sampling history-dependent behavior with adequate process coverage.
Abstract
from arXiv · showhide
We develop a new computing paradigm, which we refer to as data-driven computing, according to which calculations are carried out directly from experimental material data and pertinent constraints and conservation laws, such as compatibility and equilibrium, thus bypassing the empirical material modeling step of conventional computing altogether. Data-driven solvers seek to assign to each material point the state from a prespecified data set that is closest to satisfying the conservation laws. Equivalently, data-driven solvers aim to find the state satisfying the conservation laws that is closest to the data set. The resulting data-driven problem thus consists of the minimization of a distance function to the data set in phase space subject to constraints introduced by the conservation laws. We motivate the data-driven paradigm and investigate the performance of data-driven solvers by means of two examples of application, namely, the static equilibrium of nonlinear three-dimensional trusses and linear elasticity. In these tests, the data-driven solvers exhibit good convergence properties both with respect to the number of data points and with regard to local data assignment. The variational structure of the data-driven problem also renders it amenable to analysis. We show that, as the data set approximates increasingly closely a classical material law in phase space, the data-driven solutions converge to the classical solution. We also illustrate the robustness of data-driven solvers with respect to spatial discretization. In particular, we show that the data-driven solutions of finite-element discretizations of linear elasticity converge jointly with respect to mesh size and approximation by the data set.
1. Introduction
Data-driven computing replaces empirical material modeling with direct use of experimental data alongside conservation laws. The paper formulates this paradigm and demonstrates convergence and discretization robustness in nonlinear trusses and linear elasticity.
- Motivation: Classical computation calibrates empirical material models from observations before solving, introducing modeling error and uncertainty.These issues are especially relevant for systems with high-dimensional phase spaces and complex behavior.
- Motivation: Prior approaches generally use observational data to identify parameters or augment material models rather than replace them.Material informatics combines database methods with machine-learning regression to produce predictive quantitative models.
- Proposed paradigm: Data-driven computing formulates calculations directly from experimental material data and essential constraints, bypassing empirical material modeling.The approach preserves compatibility, equilibrium, and existing numerical discretization schemes.
- Proposed paradigm: Data-driven solvers minimize phase-space distance to a prespecified data set subject to conservation-law constraints.Equivalently, they seek a constraint-satisfying state closest to the data set.
- Validation and analysis: Tests on nonlinear three-dimensional trusses and linear elasticity show good convergence with data-point count and local data assignment.The solutions also converge to classical solutions as the data set increasingly approximates a classical material law, with joint convergence under mesh refinement and data-set approximation.
2. Truss structures
The truss solver assigns phase-space data points to bars while enforcing compatibility and equilibrium, using iterative local searches and linear solves. Numerical tests show convergence as sampling becomes finer, robustness to initialization, and slower convergence for noisy data.
- Data representation: Truss-bar material behavior is represented by local sets of strain–stress pairs in phase space, potentially differing between bars.Each bar deforms in uniaxial tension or compression, so its material state is characterized by uniaxial strain and stress.
- Variational formulation: The solver minimizes aggregate distance from the bars’ local states to their data sets subject to compatibility and equilibrium.The combined penalty function measures local departures from the corresponding data sets, while constraints enforce the truss conservation laws.
- Iterative solver: Local data assignment alternates between solving the constrained truss problem and selecting, for each bar, the nearest data point in phase space.The assignment is a local search over each data set and terminates when assignments no longer change; Voronoi cells represent the corresponding domains of influence.
- Iterative solver: The iteration remains computationally practical because each assignment step requires a linear solve whose stiffness matrix can be factorized once and reused.Subsequent iterations require inexpensive back-substitutions, despite the combinatorial complexity of assigning data points to truss members.
- Noisy-data convergence: Noisy data still produce convergence as sampling increases, but the convergence rate is roughly 1/2, lower than for noiseless data.The noisy-data experiments add Gaussian noise and evaluate convergence over 100 data sets.
- Sample quality: The local penalty decreases with data-set size but retains a spread, identifying truss members whose states are poorly covered by the available data.High penalty values indicate regions of phase space where no nearby data point exists, suggesting adaptive testing targeted at those regions.
3. Linear elasticity
Three-dimensional linear elasticity is formulated by assigning strain–stress states from local material-data sets while enforcing compatibility and equilibrium. The solver converges with increasing data resolution and retains modest iteration counts across mesh resolutions.
- Problem formulation: Linear elasticity uses twelve-dimensional phase space formed by pairs of symmetric strain and stress tensors.Each material point is characterized by a local data set of admissible strain–stress states.
- Data-driven solver: The data-driven finite-element problem minimizes a global penalty for departures from local data sets subject to compatibility and equilibrium.Compatibility is enforced through nodal displacements, while equilibrium is enforced with Lagrange multipliers.
- Material symmetries: Material symmetry reduces sampling requirements by exploiting the SO(3) orbit of isotropic strain–stress data points.For isotropic materials, rotated versions of a data point belong to the local data set, although optimal principal directions may still require solving the general optimality equations.
- Numerical tests: The tensile-specimen tests compare coarse and fine finite-element meshes with average element sizes h = 1mm and h = 0.5mm.The coarse mesh has 811 elements, while the fine three-dimensional mesh has 6,428 elements.
- Convergence: The global functional decays through local assignment iterations for both mesh resolutions, with convergence iterations remaining modest as material-data sample size increases.The result is reported for increasingly large data sets in the three-dimensional tensile-specimen problem.
- Convergence: RMS stress and strain errors show ostensibly linear convergence with increasing data resolution.Errors are plotted against the cubic root of the number of data points using reference-solution moduli.
4. Mathematical analysis of convergence
The analysis formulates data-driven mechanics in phase space as a distance minimization between material data and constraint states, then establishes convergence as data sets and discretizations become increasingly accurate.
- Abstract formulation: The data-driven problem is posed on a phase space Z with a material-data set E and a constraint set C encoding essential constraints and conservation laws.For linear elasticity, states comprise strain and stress fields; C imposes compatibility and equilibrium.
- Abstract formulation: The double-minimum formulation equivalently finds the material-data point closest to satisfying constraints or the constraint state closest to the material data.This replaces exact intersection with distance minimization when imperfect experimental data do not intersect C.
- Convergence with respect to sample size: Convergence to classical solutions is obtained when finite data sets increasingly approximate a limiting material set, under isolated intersections and a transversality condition.The assumptions include control of data-set spread and density, with transversality parameter 0 ≤ λ < 1.
- Convergence with respect to sample size: Precise convergence rates follow when the data-set density ρk and spread tk are related to the number of samples Nk.For noise-free trusses, both scale as N^-1 with α = 1; noisy trusses scale as N^-1/2 with α = 1/2; noise-free plane-stress elasticity scales as N^-3 with α = 3.
- Convergence with respect to mesh size: The data-driven solutions remain convergent under spatial discretization when data-set fidelity increases appropriately with mesh resolution.Thus, finite-element solutions converge jointly with mesh refinement and approximation of the material data.
5. Summary and concluding remarks
Data-driven computing replaces empirical material modeling with direct use of material data constrained by conservation laws. Its solvers show convergence, analytical tractability, and robustness, while their broader scope depends on the complexity of material behavior and data coverage.
- Data-driven solvers assign each material point the closest state from a prespecified data set while satisfying essential constraints and conservation laws.The objective minimizes distance to the data set in phase space subject to those constraints.
- Good convergence properties are demonstrated for nonlinear three-dimensional trusses and finite-element discretized linear-elastic solids, including local data assignment.
- The variational structure makes the problem amenable to analysis and yields convergence to classical solutions as the data set increasingly approximates a limiting constitutive law.Classical solutions are recovered when the data set coincides with the graph of a material law.
- For inelastic materials, sampling history-dependent behavior may require exceedingly large and complex data sets.The authors identify Data Science and Big Data management as potentially beneficial for handling such data sets.
- Data-driven computing enables direct links between solution results and material data, supporting traceability, data-fidelity measures, error-based data refinement, and checks of constitutive-relation suitability.