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Finite-Strain Contact Analysis

Based on Optimization Approaches

�Student: Dhyey Bhavsar

Advisor: Kurt K. Maute

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Contact Analysis Overview

Why study of contact problems?

Why optimization approach?

Continuum mechanical formulation

Large deformations, high stresses and

subsequent non-linearity in the system

Special loading, i.e, friction, impact
Friction-less and Frictional contact, Wriggers (2006)

Comparison of contact, Ansys Inc. [10]

Schematic of contact b/w two body [9]

Limitations of iterative newton methods
Derived from first the principle of potential energy minimization.
Variation of total potential energy,

First and second order optimality criteria

Boundary-value problem

w/ contact treatment

Strong Form of energy to

weak form of energy

Contact constraint mechanism
Contact detection & constraint

enforcement

Solver strategy

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How to detect contact?

How to enforce contact constraint?

Equilibrium of forces (static) ; momentum conservation (dynamic)
Friction-less: Hertz-Signorini-Moreau condition, Wriggers (2006)

Frictional: Constitutive law, i.e., coulomb’s friction model

Level-set field-based constraint formulation

Schematic of closest point projection method, Tupek et al.(2021)

Level-set based contact detection, Mosby et al. (2021)

Contact Constraint Modeling

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Contact Constraint Modeling

Optimization-based solution

methods

Explicit Contact-domain methods

Derived methods (Implicit)

Lagrange formulation,

Wriggers (2006); Mosby et al. (2021)

Penalty formulation,

Wriggers (2006); Tupek et al. (2021)

Lagrange formulation

Penalty formulation

Third medium formulation, Wriggers et al. (2013)

Third medium contact approach,

Wriggers et al. (2013)

Numerical Results b/w Penalty and Lagrange Formulation, Ansys Inc. [10]

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Algorithms for Numerical Solvers

Traditional Newton-based solver scheme

Specific solver scheme for optimization problem

Newton Solver, Kelley et al. (2003)	Struggles to capture instabilities, slow convergence rates Unstable convergence Local minima
Modified Newton Solver, Müller et al. (2008)	Adjust stiffness matrix to keep it PD May lead to sub-optimal search directions “Shift” strategy for stiffness matrix is problem dependent
Non-smooth newton solver, Macklin et al. (2019)	Uses Fischer-Burmeister function to compute complimentary error Robust but computationally expensive

Objective: Minimize energy or energy-like functional

Contact is modeled as an inequality constraint minimization

sub-problem & solved in augmented lagrangian setting

Friction-less: Constrained minimization problem

Frictional: “Nearly” constrained minimization problem

Trust-region algorithm, Tupek et al. (2021) ; Overlap removal algorithm, Mosby et al. (2021)

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Algorithms for Numerical Solvers

Traditional Newton-based solver scheme

Specific solver scheme for optimization problem

Newton Solver, Kelley et al. (2003)	Struggles to capture instabilities, slow convergence rates Unstable convergence Local minima
Modified Newton Solver, Müller et al. (2008)	Adjust stiffness matrix to keep it PD May lead to sub-optimal search directions “Shift” strategy for stiffness matrix is problem dependent
Non-smooth newton solver, Macklin et al. (2019)	Uses Fischer-Burmeister function to compute complimentary error Robust but computationally expensive

Objective: Minimize energy or energy-like functional

Contact is modeled as an inequality constraint minimization

sub-problem & solved in augmented lagrangian setting

Friction-less: Constrained minimization problem

Frictional: “Nearly” constrained minimization problem

Trust-region algorithm, Steihaug (1983), Toint (1988) ; Contact Constraint Optimization, Tupek et al. (2021); Overlap removal algorithm, Mosby et al. (2021)

Strain-energy stored

Work done by

External force

Work done by

Surface traction

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Algorithms for Numerical Solvers

Trust-region algorithm, Tupek et al. (2021)

Original Unconstrained

minimization problem

Approximate

sub-problem

Minimize sub-problem

with constraint

What if the problem is highly non-convex?

What if minimum solution is outside the current trust-region?

Trust-region algorithm uses negative curvature information & project onto Trust-region boundary, indicating direction of minimum

Project update step back to trust-region boundary and return corresponding solution to the projected step as an approximate one

Schematic of Trust-region algorithm, Hofer (2022)

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References

[1] Peter Wriggers. Computational Contact mechanics, volume 2. American Institute of Physics Inc., 2006.

[2] Peter Wriggers and C. Miehe. Contact constraints within coupled thermomechanical analysis—a finite element model.

Computer Methods in Applied Mechanics and Engineering, 113:301–319, 1994.

[3] P. Wriggers, J. Schroder, and A. Schwarz. A finite element method for contact using a third medium. Computational

Mechanics, 52(4):837–847, October 2013.

[4] Carl T Kelley. Solving nonlinear equations with Newton’s method. SIAM, 2003.

[5] Michael Muller. Postbuckling analysis stabilized by penalty springs and intermediate corrections. Computational

Mechanics, 42:631–654, 2008.

[6] Miles Macklin, Kenny Erleben, Matthias Müller, Nuttapong Chentanez, Stefan Jeschke, and Viktor Makoviy-chuk. Non-

smooth newton methods for deformable multi-body dynamics. ACM Transactions on Graphics, 38(5):1–20, oct 2019.

[7] Michael Tupek and Brandon Talamini. Optimization-based algorithms for nonlinear mechanics and frictional contact. 9

2021.

[8] Matthew Mosby, Michael Tupek, and Johnathan Vo. A simple levelset contact algorithm for large overlap removal and

robust preloads. 9 2021.

[9] Webpage: Contact Mechanics Model, EPFL

[10] Webpage: Introduction to Contact Mechanics, Ansys Inc.