Source-linked AI summary
Data-Driven Computing in Dynamics
Trenton Kirchdoerfer, Michael Ortiz
TL;DR
The paper addresses how Data Driven Computing can incorporate time integration without relying on prespecified material models. It formulates time-dependent distance-minimizing and maximum-entropy solvers using free-energy minimization under compatibility and time-discretized momentum constraints, then evaluates their convergence. Both solver types converge as data sets approach an underlying model, while max-ent solvers also handle outliers and fixed-distribution data and are more effective for step-driven transients.
Problem
Scientific computing needs to use increasingly rich material data directly because prespecified modeling assumptions can restrict reproduction of observed behavior.
Method
The paper extends distance-minimizing and maximum-entropy Data Driven solvers to dynamics through free-energy minimization over phase space with compatibility and time-discretized momentum constraints.
Results
Both distance-minimizing and max-ent solutions converge as data sets approach an underlying model, while max-ent solutions are robust to outliers and converge toward fixed distributions.
Takeaways & Limitations
Max-ent solvers additionally show greater efficacy for step-driven transient solutions through accumulated improvements of solutions over time.
Takeaways & Limitations
The implementation leaves efficiency improvements for annealing schedules and time integration unaddressed, including data summarization, range searches, radial cutoffs, and improved initialization.
Abstract
from arXiv · showhide
We formulate extensions to Data Driven Computing for both distance minimizing and entropy maximizing schemes to incorporate time integration. Previous works focused on formulating both types of solvers in the presence of static equilibrium constraints. Here formulations assign data points a variable relevance depending on distance to the solution and on maximum-entropy weighting, with distance minimizing schemes discussed as a special case. The resulting schemes consist of the minimization of a suitably-defined free energy over phase space subject to compatibility and a time-discretized momentum conservation constraint. The present selected numerical tests that establish the convergence properties of both types of Data Driven solvers and solutions.
1 Introduction
The paper motivates scientific computing that operates directly on rich material data rather than relying on restrictive constitutive models. It extends Data Driven Computing to dynamics and tests convergence of distance-minimizing and entropy-maximizing solvers.
- Large data sets can exceed the richness of empirical summaries, while model assumptions restrict reproduction of observed behavior.
- Material Informatics processes material data but lacks explicit acknowledgement of physics field equations constraining material behavior.
- Parameter-identification methods calibrate prespecified material laws, whereas Data Driven Computing uses fundamental material data directly without material models.
- Earlier Data Driven Computing reformulated mechanics using compatibility and conservation laws as material-independent constraints and selected the constraint-manifold point closest to the data set.
- The present work extends Data Driven Computing to dynamics using variational time stepping, with distance-minimizing methods treated as a special annealing case.
- Numerical tests assess convergence of max-ent and distance-minimizing solutions, including max-ent behavior for random material data.
2 Review of Data Driven schemes
Data Driven schemes seek constraint-compatible solutions close to discrete material data, while maximum-entropy weighting handles finite data bands and outliers. The max-ent formulation minimizes free energy, and simulated annealing addresses its non-convexity.
- Data Driven formulation: Discrete material data may not intersect the compatibility, equilibrium, and boundary constraint set, requiring relaxation that minimizes deviation from the data while satisfying constraints.
- Data clustering: Distance-minimizing solvers converge for noisy data only when data sets approach a phase-space graph, motivating probabilistic relevance weighting for finite data bands.
- Data clustering: Maximum-entropy weighting combines unbiased relevance with reduced weights for points farther from the phase-space solution through a Pareto weight β ≥0.
- Max-ent solver: The max-ent solver minimizes free energy over the constraint set, with closest-point projection appearing in its optimality and fixed-point formulations.
- Data clustering: For finite β, every material data point influences the solution, but influence decreases with distance to the solution.
- Data clustering: An outlier marginally closer to the constraint set does not significantly alter the solution when a large cluster of material data points is present.
- Simulated annealing: Strong non-convexity creates multiple wells around data points, so simulated annealing evolves reciprocal temperature alongside fixed-point iteration.
- Simulated annealing: Annealing schedules select β updates to ensure local contractivity, starting from a sufficiently small reciprocal temperature and a convexifying β0.
3 Application to dynamics
The dynamics formulation applies Data Driven Computing to truss structures by combining phase-space material data with compatibility, equilibrium, and time-discretized constraints. Newmark time integration reduces the dynamic equations to an equivalent static problem at each step.
- Truss phase space: Truss members are modeled through local data sets of strain–stress pairs, with the global data set formed as their Cartesian product.Each bar’s phase-space state is ze = (εe, σe), representing uniaxial strain and stress.
- Dynamic constraints: At time tk, the truss state is constrained by compatibility and equilibrium equations involving nodal displacements, accelerations, applied forces, member geometry, member volumes, and the mass matrix.The matrices Be encode member geometry and connectivity, while M represents inertia.
- Time integration: The equations are integrated in time by discretizing displacement, velocity, and acceleration with the Newmark algorithm.The Newmark parameters β and γ enter the time-discrete formulation.
- Time integration: Newmark predictors transform the dynamic constraints into an equivalent static problem, followed by corresponding constraint updates.This enables the dynamic Data Driven problem to be treated through successive time-discrete equilibrium formulations.
- Distance formulation: The truss phase spaces are metrized using Euclidean distances, allowing distances between local states to be used in the Data Driven formulation.A global norm and associated distance are also introduced for truss states.
4 Numerical tests
Numerical tests assess dynamic Data Driven solvers on a 1,246-member truss using noisy and randomly distributed material data. They compare distance-minimizing and maximum-entropy schemes through convergence errors, selection frequencies, and displacement histories.
- Test case: The truss test contains 1,246 members and uses a 10-cycle sine excitation resolved over 300 time steps.A nonlinear stress-strain relation and consistently integrated Newton-Raphson solution provide the reference response.
- Error metric: The error metric compares time-integration solutions across multiple timesteps without overweighting long-time phase error.Integration begins after the first time step because initial displacement discrepancies could otherwise create a singularity.
- Uniform convergence: Distance-minimizing solutions converge as N^-1/2, while maximum-entropy solutions converge linearly for capped noisy data sets.The convergence assessment uses histograms compiled from 30 randomly generated material-set samples at each data-set size.
- Random material behavior: For fixed-distribution random material behavior, the mean error of sampled maximum-entropy solutions converges to zero with data-set size at a rate of approximately 0.19.The error histograms contain 30 randomly generated data-set samples for every data-set size, and the slower rate is associated with wider data spread as a possible explanation.
- General performance: Maximum-entropy solutions improve on distance-minimizing solutions even for noiseless data and maintain fidelity to the reference displacement history.Distance-minimizing selection can favor outliers, whereas the maximum-entropy solver is described as robust to noisy inputs through clustering.
5 Summary and discussion
The paper distinguishes model-independent, reusable material data from model-specific repositories while demonstrating convergence of transient Data Driven solvers. It also identifies unimplemented algorithmic and time-integration improvements that could enhance numerical performance.
- Both distance-minimizing and max-ent Data Driven solutions converge as data-set sequences approach an underlying model.
- Max-ent solutions are robust to outliers and converge when data-set sequences approach a fixed distribution.
- Max-ent solvers are more effective for step-driven transient solutions because solution improvements accumulate over time.
- Fundamental data are determined by compatibility and conservation laws rather than prespecified material models.This model-independent representation distinguishes Data Driven Computing from repositories containing parametric, model-specific data.
- Fundamental material data can be reused across purposes, blended across sources, and stored in publicly editable repositories.
- The implementation leaves data summarization, efficient range searches, radial cutoffs, and time-step-informed annealing initialization unaddressed.The authors state that these improvements could dramatically improve numerical performance.