Towards Digital Twin for Integrated High-Volume Manufacturing and Product Performance

Jacob Fish

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Carleton Professor

Department of Civil Engineering & Engineering Mechanics

Director, initiative for Computational Science & Engineering

Columbia University

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Collaborators: Andrea La Spina, Yang Yu, Andrew Beel, Junhe Cui (Columbia University), Jaan Simon (RWTH Aachen), Zifeng Yuan (Peking University), Venkat Aitharaju (General Motors)

Digital Twin for Automotive Industry

High-Volume Lightweight (Carbon Fiber) Manufacturing

Product Design

1. Computational Complexity

Computational Challenges

2. Scale mixing

Outline

• Computational Challenges
• Data-Physics Driven Multiscale Approach for High-Pressure Resin Transfer Molding
• Single-Phase Multiscale Flow Model: Saturated Pressure-Dependent
• Multiphase-Multiscale Flow Model: Momentum Exchange, Unsaturated Flow Based on Phase Field & Capillary Pressure
• Data-Physics Driven Reduced Order Homogenization (dpROH) for component analysis
• Addressing scale mixing in hot spots: Scale-Separation-Free Approach
• Coupled Chemo-Thermo-Mechanical Reduced-Order Multiscale Model for Predicting Micromechanical Residual Stresses and Distortions
• Conclusions

Data-Physics Driven Multiscale Approach for High-Pressure Resin Transfer Molding

J. Cui, A. La Spina, J. Fish. Data-Physics Driven Multiscale Approach for High-Pressure Resin Transfer Molding (HP-RTM) Computer Methods in Applied Mechanics and Engineering, in print.

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J. Cui, A. La Spina, J. Fish. Data-Physics Driven Multiscale Approach for High-Pressure Resin Transfer Molding in Multi-Porous Medium, in preparation.

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A. La Spina, J. Fish. Data-Physics Driven Multiscale Approach for Multi-Phase Fluid Flow in Porous Media via Space-Time Computational Homogenization, in preparation

HP-RTM molded 1.5-meter-by-0.5-meter CFRP rib in 20 minutes with 60-percent fiber volume and less than 2 percent voids

HP-RTM molded BMW i8 side-frame

Single-phase saturated flow model�Two-scale composite

Step 1: Define two scales

Step 2: Rescaling by normalization

Step 3: Asymptotic expansion

Low pressure (classical):

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High pressure:

Single-phase saturated flow model�Two-scale composite

Step 4: Various order governing equations

Pressure-Dependent Average Velocity

Pressure-Dependent Instantaneous Permeability

Surrogate Model Verification

Training dataset:

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• 7500 data points
• (-50, 350) bar/m
• Time: 5 minutes
• Accuracy: 99.9%

Single-phase saturated flow model �Three-Scale Model

Step 1: Define three scales

Step 2 and Step 3 are similar to two scale fibrous composite

Single-phase saturated flow model �Three-Scale Model

Microscale RVE

Mesoscale RVE

Macroscale

Woven Composite RVE

Solve

Navier-Stokes-Brinkman equation

Too expensive!

Artificial neural network based surrogate model is trained to avoid the high computational cost

Training dataset:

• 15625 data points
• (-50, 200) bar/m
• Training time: 2 hours
• Accuracy: > 99%

Components of the permeability K

Surrogate Model Verification

Verification against DNS�Three-scale 2D model problem

Direct numerical simulation

Nonlinear 3-scale homogenization

Mold fill-in time comparison

Mold filling -3D model

Mold fill-in time comparison

Saturation issue

Re = 1.5

Re = 30

Re = 0.3

Significant unsaturated RVEs

Negligible unsaturated RVEs

Re increases

Multiphase-Multiscale Approach*�Based on Phase Field and Capillary Pressure

Current approaches:

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• Ignore saturation effects or represent it empirically
• Do not account for hysteresis effect

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Proposed approach:

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• Homogenization of two-phase flow in porous medium from pore to component scale using phase field approach
• Determine saturation-dependent parameters based on underlying microstructure
• Gated recurrent units (GRUs) neural networks to efficiently link micro and macro scales

*Also applicable to Vacuum-Assisted RTM

Breakdown of Scale Separation�Homogenization Error Indicator (EI)

• Based on the normalized value of the terms neglected in asymptotic expansion

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• Employ composite grid method combining nonlinear homogenization with DNS where

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Composite Grid Flow Solver

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Data-Physics Driven Reduced Order Homogenization (dpROH) for component analysis

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J. Fish. Practical Multiscaling, Wiley, 2013.

J. Fish, Y. Yu. Data‐physics driven reduced order homogenization. International Journal for Numerical Methods in Engineering 124 (7), 1620-1645, 2023.

Y. Yu, J. Fish. Data-Physics Driven Reduced Order Homogenization for Continuum Damage Mechanics at Multiple Scales, International Journal for Multiscale Computational Engineering 22 (1), 2023.

Physics Based Reduced Order Homogenization (pROH)

Physics Based Reduced Order Homogenization (cont)

Physics Based Reduced Order Homogenization (cont)

Locking of One-Partition-per-Phase Model in Matrix Dominated Mode of Deformation

Perfectly plastic

Elastic

Since the eigenstrain in the matrix is assumed to be constant in one-partition-per-phase model, the elastic inclusion is forced to evolve from the round to oval shape.

in matrix dominated mode of deformation

Data-Physics Driven Reduced Order Homogenization (cont)

100 🡪 6

144 🡪 8

Data-Physics Driven Reduced Order Homogenization (cont)

Validation: Quasi-isotropic plate with a hole under simple shear problem

Error definition:

Critical element error

Geometry & BCs:

Critical element

Von mises stress distribution

Reference Solution

Data-Physics Driven Reduced Order Homogenization (cont)

Validation: Quasi-isotropic plate with a hole under three-point bending problem

Critical element error

Geometry & BCs

Error definition:

Von mises stress distribution

Reference Solution

Critical element

Data-Physics Driven Reduced Order Homogenization (cont)

Extension to Scale Mixing

Computational Continua (C²)

Nonlocal Quadrature

Physical domain:

Parent domain:

Coarse-Scale Problem

Integration by parts and applying nonlocal quadrature:

Coarse-Scale Weak Form:

Coarse-Scale Weak Form:

Model Verification

Unit cells

Unstructured mesh

Attractive application for �waffle, ribbed, hollow-core plates

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Coupled Chemo-Thermo-Mechanical Reduced-Order Multiscale Model for Predicting Micromechanical Residual Stresses and Distortions

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Z. Yuan, S. Felder, S. Reese, J.W. Simon, J. Fish. A coupled thermo-chemo-mechanical reduced-order multiscale model for predicting residual stresses in fibre reinforced semi-crystalline polymer composites. International Journal for Multiscale Computational Engineering Vol. 18(5), pp. 519-546, (2020)

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Z. Yuan, V. Aitharaju, J. Fish. A coupled thermo‐chemo‐mechanical reduced‐order multiscale model for predicting process‐induced distortions, residual stresses, and strength. International Journal for Numerical Methods in Engineering, Vol. 121(7), pp. 1440-1455, (2020)

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Nylon 6 (semicrystalline polyamide)

Illustration of the coupling phenomena between total degree of crystallinity χ, temperature field θ, and displacement field u, as well as illustration of the modeling strategy and the assumption of two successive processes I and II.

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Noteworthy, coupling effects highlighted in yellow arrows are captured by the proposed theory, whereas coupling phenomena depicted in grey arrows are

assumed to have only minor influences and are thus neglected.

Multiscale Chemo-Thermo-Mechanical Model

Multiscale Chemo-Thermo-Mechanical Model�Thermoset polymer composite

a, coupling of chemo-thermo-mechanical processes at spatial multiple scales, b, model reduction and math-based upscaling, c, predicted residual stresses induced by manufacturing, d, model validation at a component level

Composite Underbody Assembly Design

Comparison of crushed steel (64kg) and carbon fiber (48kg) underbody assembly

In collaboration with General Motors

Conclusions

• The data-physics driven multiscale flow model can reasonably well match the direct Navier Stokes solver in a medium to high pressure regime with “little” overhead in comparison to the classical Darcy-Stokes solver.
• Multiphase-multiscale flow model is necessary to account for momentum exchange, hysteresis/history, and capillary pressure effects.
• Multiscale model of cure/crystallization kinetics is necessary to resolve micromechanical residual stresses affecting component strength.
• Scale-separation-free multiscale method with relatively little overhead over the classical computational homogenization methods is necessary in the vicinity of hot spots.
• Prediction: commercial grade data-physics-driven multiscale software for integrated process–performance design along the lines presented here could be operational in 5 to 7 years.

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Computational Science and Engineering @ Columbia University

iCSE Collaborative Research areas

• Hybrid Physics-Data Driven Computational Science and Engineering
• Computational Mathematics and Numerical Analysis
• Parallel Computation and Petascale/Exascale Computing
• Uncertainty Quantification and Stochastic Modeling
• Reduced Order Modeling
• Inference, Verification and Validation of Models
• Multiphysics and Multiscale Modeling
• Computational Climate
• Computational Biomechanics and Biology
• Computational Neuroscience
• Computational Imaging
• Computational Health
• Computational Mechanics and Materials
• Computational Chemistry
• Graphics, Animation and Visualization
• Optimization
• Computational Design, Decision-Making, and Finance

iCSE Members and Affiliates Program (MAP)