(Hyper-reduced order methods and domain decomposition procedures for efficiently modeling non-linear geometric structural behavior)
Author: Raul Rubio Serrano
Supervisors
Joaquín Hernández Ortega
Àlex Ferrer Ferré
Modeling lattice structures via reduced order models
Motivation & Objectives
Empirical Interscale Finite Element Method (EIFEM)
Using EIFEM as preconditioner (Objective 2)
Non-linear regime (Objective 3)
Integrating EIFEM 1D elements in stardard FE codes (Objective 1)
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Metamaterials
Architected materials��
Optimized structures
�
Applications
Metamaterials
Architected materials��
Optimized structures
�
Applications
Lightweight structures
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Metamaterials
Architected materials��
Optimized structures
�
Applications
Lightweight structures
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Metamaterials
Architected materials��
Optimized structures
�
Applications
Lightweight structures
Energy absortion
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Metamaterials
Architected materials��
Optimized structures
�
Applications
Lightweight structures
Energy absortion
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Metamaterials
Architected materials��
Optimized structures
�
Applications
Lightweight structures
Energy absortion
Biomedical devices
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Metamaterials
Architected materials��
Optimized structures
�
Applications
Lightweight structures
Energy absortion
Biomedical devices
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EIFEM
Integrating 1D EIFEM elements in standard codes (bROM)
Using EIFEM as a preconditioner
Expanding EIFEM to large strains (non-linearities)
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Objectives
EIFEM
Integrating 1D EIFEM elements in standard codes (bROM)
Using EIFEM as a preconditioner
Expanding EIFEM to large strains (non-linearities)
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Objectives
EIFEM
J.A. Hernández, A. Giuliodori, E. Soudah, Empirical Interscale Finite Element Method (EIFEM) for modeling heterogeneous structures via localized hyperreduction. 2024.
J.A. Hernandez, A multiscale method for periodic structures using domain decomposition and ecm- hyperreduction. 2020
A. Giuliodori, J.A Hernández, E. Soudah, Multiscale modeling of prismatic heterogeneous structures based on a localized hyperreduced-order method. 2023
Empirical Interscale Finite Element Method (EIFEM)
EMPIRICAL
INTERSCALE
FEM
J.A. Hernández, A. Giuliodori, E. Soudah, Empirical Interscale Finite Element Method (EIFEM) for modeling heterogeneous structures via localized hyperreduction. 2024.
Downscaling
Upscaling
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EIFEM
Training via full-scale FE analysis
Coarse finite element
EMPIRICAL
INTERSCALE
FEM
Training via full-scale FE analysis
EIFEM
J.A. Hernández, A. Giuliodori, E. Soudah, Empirical Interscale Finite Element Method (EIFEM) for modeling heterogeneous structures via localized hyperreduction. 2024.
Empirical Interscale Finite Element Method (EIFEM)
Downscaling
Upscaling
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Coarse finite element
LLM
Fictitious frames
Subdomains
Three field formulation
d
K. C. Park, C. A. Felippa, U. A. Gumaste, A localized version of the method of Lagrange multipliers and its applications. 2000.
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EIFEM
d
LLM
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EIFEM
d
LLM
EIFEM
rb
def
rb
def
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d
LLM
EIFEM
rb
def
rb
def
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EIFEM
Store displacements
Store Lagrange multipliers
Singular value decomposition
Solve with FE
d
EIFEM
1)
2)
3)
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d
EIFEM
1)
2)
3)
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EIFEM
FIBRE composite manufacturing technologies FOR the automation and modular construction in shipYARDS - FIBRE4YARDS (Grant agreement 101006860); Funding agency: European Comission; Program: H2020�Task: Simulation of pultruded beams using reduced order models. As a result of this task the beam Reduced Order Model (bROM) was developed.
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FIBRE4YARDS
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bROM: motivation
Simulation of beams made of composite materials
(Or 1D like structures)
Isotropic materials
Orthotropic materials
bROM: new beam model for isotropic and orthotropic materials
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bROM: motivation
Isotropic materials
Orthotropic materials
bROM: new beam model for isotropic and orthotropic materials
Linear elasticity
Simulation of beams made of composite materials
(Or 1D like structures)
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bROM: motivation
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bROM: motivation
FE
EIFEM
160.000 elements
200 elements
7 DOFs
EIFEM
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bROM: EIFEM
1
Condensation
2
Interpolation
3
EIFEM
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bROM: condensation
1
Condensation
2
Interpolation
3
Perturbations
EIFEM
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bROM: regression
1
Condensation
2
Interpolation
3
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bROM: validation
R.Rubio, A.Ferrer, J.A. Hernandez, X.Martinez, bROM: an extension of beam theory through model order reduction. 2024
bROM: an extension of beam theory through model order reduction
Journal: Computers and Structures
Impact index: 9.0
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bROM: outcome
Analysis of composite beams using model order reduction
In proceedings: Revista de materiales compuestos
Location: Gijón, Spain
Date: June 2023
EIFEM
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Timeline: WP1
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Preconditioner: Motivation
Motivation
Direct solver
Memory usage
�
Computational time
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Motivation
Iterative solvers + preconditioning
Direct solver
Memory usage
�
Computational time
Memory usage
�
Computational time
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Motivation
Iterative solvers + preconditioning
Reduced order modeling (ROM)
Direct solver
Memory usage
�
Computational time
Memory usage
�
Computational time
Memory usage
Computational time
Speed-accuracy trade-off
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Motivation
Iterative solvers + preconditioning
Reduced order modeling (ROM)
Use ROM as preconditioner within an iterative solver
Direct solver
Memory usage
�
Computational time
Memory usage
�
Computational time
Memory usage
Computational time
Speed-accuracy trade-off
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Domain decomp.
Multigrid
Domain decomp. + ROM
Dirichlet-Neumann
Neumann-Neumann
Balancing Neumann-Neumann
FETI / FETI-DP
BDDC
Windlund & Tosselli 2006
Mandel 1993
Hirschler et al. 2023
Klawonn et al. 2024
Trottenberg et al. 2001
Preconditioners: State of the art
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Domain decomp.
Multigrid
Domain decomp. + ROM
Dirichlet-Neumann
Neumann-Neumann
Balancing Neumann-Neumann
FETI / FETI-DP
BDDC
Windlund & Tosselli 2006
Mandel 1993
Hirschler et al. 2023
Klawonn et al. 2024
Trottenberg et al. 2001
Preconditioners: State of the art
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Preconditioning
Use EIFEM as preconditioner within an iterative solver
Richardson iteration
P is called preconditioner
Jacobi P = Diag(A)
Gauss-Seidel P = L(A)
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Preconditioning
Use EIFEM as preconditioner within an iterative solver
Richardson iteration
P is called preconditioner
Jacobi P = Diag(A)
Gauss-Seidel P = L(A)
Dirichlet-Neumann
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Preconditioning
Use EIFEM as preconditioner within an iterative solver
Richardson iteration
P is called preconditioner
Jacobi P = Diag(A)
Gauss-Seidel P = L(A)
Dirichlet-Neumann
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Preconditioning
Use EIFEM as preconditioner within an iterative solver
Richardson iteration
P is called preconditioner
Jacobi P = Diag(A)
Gauss-Seidel P = L(A)
Dirichlet-Neumann
Use EIFEM as preconditioner within an iterative solver
Dirichlet-Neumann
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Richardson iteration
P is called preconditioner
Jacobi P = Diag(A)
Gauss-Seidel P = L(A)
Preconditioning
Use EIFEM as preconditioner within an iterative solver
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Dirichlet-Neumann
Richardson iteration
P is called preconditioner
Jacobi P = Diag(A)
Gauss-Seidel P = L(A)
Preconditioning
Use EIFEM as preconditioner within an iterative solver
Neumann-Neumann
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Richardson iteration
P is called preconditioner
Jacobi P = Diag(A)
Gauss-Seidel P = L(A)
Preconditioning
Use EIFEM as preconditioner within an iterative solver
Neumann-Neumann
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Richardson iteration
P is called preconditioner
Jacobi P = Diag(A)
Gauss-Seidel P = L(A)
Preconditioning
Use EIFEM as preconditioner within an iterative solver
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Neumann-Neumann
Richardson iteration
P is called preconditioner
Jacobi P = Diag(A)
Gauss-Seidel P = L(A)
Preconditioning
Preconditioning with EIFEM
Use EIFEM as preconditioner within an iterative solver
Very fast
inverse!!
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Richardson iteration
P is called preconditioner
Jacobi P = Diag(A)
Gauss-Seidel P = L(A)
CG
Compute residual
R = b-Axn
Compute α, β
Update solution
1
3
xn+1 = xn - α(b-Axn) + β(xn-xn-1)
2
Preconditioning with EIFEM
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PCG
Compute residual
Update residual
R' = R-xEIFEM
Update solution
1
2
3
Solve precon.
1a
Compute α, β
1b
R = b-Axn
xn+1 = xn - αR'+ β(xn-xn-1)
PCG
Compute residual
Update residual
R' = R-xEIFEM
Update solution
1
2
3
Solve precon.
1a
Compute α, β
1b
R = b-Axn
xn+1 = xn - αR'+ β(xn-xn-1)
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Preconditioning with EIFEM
PCG
Compute residual
Update residual
R' = R-xEIFEM
Update solution
1
2
3
Solve precon.
1a
Compute α, β
1b
R = b-Axn
xn+1 = xn - αR'+ β(xn-xn-1)
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Preconditioning with EIFEM
EIFE Q4 element
Case study
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Case study
Training configuration 1
Training configuration 2
EIFE Q4 element
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CASE STUDY
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Case studies
Training configuration 1
CG 1018 iterations
CG + EIFEM 314 iterations
CG + EIFEM 3.24x faster convergence
CASE STUDY
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Case studies
Training configuration 2
CG 1018 iterations
CG + EIFEM 233 iterations
CG + EIFEM 4.36x faster convergence
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Case studies
CASE STUDY
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Case studies
Iteration 3
Iteration 10
Preconditioning iterative solvers via empirical interscale finite element method (EIFEM)
Planned publication
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Preconditioner: outcome
Solving lattice structures via iterative solvers and �reduced-order models
9th European Congress on Computational Methods in Applied Sciences and Engineering
Location: Lisbon, Portugal
Date: June 2024
Preconditioning iterative solvers via reduced-order modelling for the simulation of lattice structures�Congress on Numerical Methods in Engineering
Location: Aveiro, Portugal
Date: September 2024
EIFEM
Timeline: WP2 & WP3
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Non-linearities: motivation
Many real life applications endure large strains (or rotations)
Solve every step
Computationally very demanding
H. Yang, L. Ma, 1D to 3D multi-stable architected materials with zero Poisson's ratio and controllable thermal expansion, Materials & Design 2020
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Non-linearities: motivation
ROM
Iterative solver
Many real life applications endure large strains (or rotations)
Solve every step
Computationally very demanding
H. Yang, L. Ma, 1D to 3D multi-stable architected materials with zero Poisson's ratio and controllable thermal expansion, Materials & Design 2020
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Non-linearities: training
Oversampling
Periodic conditions
C. Maruccio, L. De Lorenzis, Numerical homogenization of piezoelectric textiles with electrospun fibers for energy harvesting. Frattura ed Integrità Strutturale. 2014
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Non-linearities: Coarse displacements
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Non-linearities: Coarse displacements
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Non-linearities: Coarse displacements
Modeling non-linear deformations via iterative solvers and reduced order models in lattice structures.
Planned publication
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Non-linearities: outcome
Conferences. (To be decided)
29th International Conference on Domain Decomposition Methods. Milan, June 2025
VIII International Workshop on Reduced Basis, POD or PGD-Based Model Reduction Technique (MORTech25). Paris
Timeline
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Doctoral stay
École Polytechnique Fédérale de Lausanne
Department: Chair of Numerical Modelling and Simulation
Professor: Pablo Antolin
Period: 5-6 months
Reduced Order Modeling based Inexact FETI-DP solver for lattice structures
Master’s thesis: study of the design of superelements
Student : Gerard Villalta
Co-director
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Teaching activities
Bachelor’s thesis: study of iterative solvers for structural analysis
Student : Mario Fernández
Co-director
Computational aerospace engineering
Degree in aerospace technologies
ESEIAAT (Terrassa)
bROM: an extension of beam theory through�model order reduction
Journal: Computers and Structures
Impact index: 9.0
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Articles & conferences
Analysis of composite beams using model order reduction
In proceedings: Revista de materiales compuestos
Location: Gijón, Spain
Date: June 2023
Preconditioning iterative solvers via empirical �interscale finite element method (EIFEM)
Planned publication
Solving lattice structures via iterative solvers and �reduced-order models
9th European Congress on Computational Methods in �Applied Sciences and Engineering
Location: Lisbon, Portugal
Date: June 2024
Preconditioning iterative solvers via reduced-order�modelling for the simulation of lattice structures�Congress on Numerical Methods in Engineering
Location: Aveiro, Portugal
Date: September 2024
Modeling non-linear deformations via iterative�solvers and reduced order models in lattice structures.
Planned publication
Thank you for your attention
FPU-UPC
Assemble elemental matrix and apply perturbations
Apply perturbations to rigid body modes
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bROM: condensation
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bROM: regression
Linear interpolation with local support
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Novelty
Empirical Interscale Finite Element Method (EIFEM)
Very good and fast approximation
Speed-accuracy trade-off
A. Giuliodori, J.A. Hernández, E. Soudah, Empirical Interscale Finite Element Method (EIFEM) for modeling heterogeneous structures via localized hyperreduction. 2023.
J.A. Hernández, A. Giuliodori, E. Soudah, Multiscale modeling of prismatic heterogeneous structures based on hyperreduced-order method. 2023.
J.A. Hernández, A multiscale method for periodic structures using domain decomposition and ECM-hyperreduction. 2020.
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Novelty
Empirical Interscale Finite Element Method (EIFEM)
Very good and fast approximation
Speed-accuracy trade-off
Use EIFEM as preconditioner within an iterative solver
Target full solution
Speed up full solution
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Preconditioning with EIFEM
Use EIFEM as preconditioner within an iterative solver
EIFEM