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Nonsmooth and Nonconvex Mechanics:

Challenging Problems and Solutions!

The field of nonsmooth and nonconvex mechanics is a newly developed multi-interdisciplinary research area, which involves a powerful combination of theoretical analysis in mathematical modelling of natural systems, finite deformation theory, modern physics, material science, nonlinear partial differential equations,  global optimization,  variational methods, dynamical systems, global optimization, numerical algorithms  and scientific computations. This field has undergone very considerable development in a remarkably short time. 

Nonsmooth mechanics is a research topic that integrates fundamental work in nonsmooth analysis, nonlinear partial differential equations, finite deformation theory, high pressure/high rate response, damage and failure mechanics, and computational mathematics. 

    Frictional Mechanics

Modern composite materials

Damage and fracture mechanics

Penetration mechanics

Nonconvex phenomena arise from large deformation mechanics, chaotic dynamical systems, buckling analysis and phase transitions, where the total potential energies are usually nonconvex. For example, the well-known van de Waals double-well potential is a nonconvex fourth order polynomial:

W(u) = ( u^2 - a )^2

In post-buckling of large deformed beam problem, each potential well represents one possible buckling (stable) state, while the local maximize represents the un-buckled (unstable) state. In phase transitions of solids, each potential well represents certain phase of alloys. These potential wells are very sensitive to the parameter a , input force, etc. Different numerical methods will produce totally differential results. In dynamical systems, this nonconvexity of potential leads to the so-called chaos. Numerical discretization of the nonconvex minimal potential variational problem leads to a global optimization problem, which is usually NP-Hard.

Nonconvex mechanics cover many research fields of applications, such as:

Phase transitions Hysteresis modeling Chaotic dynamical systems Buckling analysis Superconductivity Yang-Mills theory and gerenal relativity Sine-Gordon Equation and Soliton

Some well-known nonconvex variational problems:

Min P(u) = ½ (u, Au) – (u, f ) + W(u),

W = ò ½ (½ u² - a)²

The criticality condition d P(u) =0 leads to a general semi-linear nonconvex equation:

Au + d W(u) = f ,

1. Chaotic dynamics: A= - d² /dt² Þ Duffing equation:

- u_tt + ( ½ u² - a) u = f

Numerical simulation by Matlab ode23	Numerical simulation by Matlab ode15s

2. Superconductivity: A = - D Þ Landau-Ginzburg equation

- D u + ( ½ u^2 - a) u = f

Bright field images of domain states in PMN-PT. (a) PMN-PT 60/40 on the FEt side of the MPB, (b) PMN-PT 65/35

3. Liquid crystal: A = D + curl curl Þ Cahn-Hillard equation

D u + curl curl u + ( ½ u² - a) u = f

4. Nonlinear Schrödinger: A= - d² /dt² + D Þ

- u_,tt + D u + ( ½ u² - a) u = f

Traditional direct approaches for solving these nonconvex semi-linear equations are fundamentally difficult. However, by using the canonical duality theory, these equations can be solved in FINITE DIMENSIONAL space. (click here to see a review article) also here for canonical dual finite element method for solving Landau-Ginzburg equation.

Complete Solutions to Phase Transitions in Ericksen’s Bar (Gao & Ogden, 2008)

Nonconvex variational problem:

(P): min P(u) = ò [ ½ (½ u_x2 - a)² – u f ] dx

Criticality condition leads to a nonlinear differential equation

-[( u_x2 – a ) u_x ] _x = f with boundary conditions.

For a given parameter a and input f(x) , this nonlinear equation has at most three solutions at EACH material point x . To determine which solution is a global minimize of the variational problem (P) is fundamentally difficult in direct analysis. By the canonical duality theory, the canonical dual problems is

(P^d): max P ^d (s ) = - ò [½t ² s ^-1 + ½ s ² + s ]dx

s.t. - t _x = f

For a given input f(x), the linear equilibrium constraint can be solved by t(x) = ò _x f(t)dt

Then the criticality condition of this dual problem leads to an ALGEBRAIC Equation:

2s ^2 (s + 1) = t ^2(x)

In s-t space, the graph of this equation is the so-called singular elliptic curve.

For a given t (x) at each position x , this equation can be solved analytically to obtain three solutions:

s₃ ≤ s₂ ≤ 0 ≤ s₁

Therefore, the complete set of analytical solutions can be given by

u_i = - ò_x t (t) s_i ^-1 dt i =1, 2, 3

Theorem: For a given distributed input f(x), the problem (P) has at most three solutions at each position x.

u₁ (x) is a global minimize, u₂(x) is a local minimize, u₃ (x) is a local maximizer, and

P(u_i )= P^d (s_i) i =1, 2, 3

If t (x) crosses t = 0 at x=x_o, then the nonconvex variational problem (P) has nonsmooth global and local minimal solutions, i.e. u₁ (x) and u₂ (x) are nonsmooth at x_o.

Example 1: Nonsmooth Global Minimal Solution

Input force field f(x) Nonconvex total potential at each x=0.1 (movie for all x)

Nonsmooth solutions u_i dual solutions s_i

The smooth solution (yellow) is not an extremal solution !!!

The global and local minimal solutions are usually nonsmooth and cannot be captured by any Newton-type direct numerical method!!!

Tri-Duality: Nonconvex potential (dotted) and dual potential (solid) for Ericksen’s bar

This example explained the reason why different numerical methods produce differential results.

In nonconvex dynamical systems, the multiple local solutions (nonconvexity) lead to the so-called chaos!

(See the paper with Prof. Ray Ogden for details).

New book on Nonsmooth and Nonconvex Mechanics:
Nonconvex/Nonsmooth Mechanics:  Modeling, Analysis and Numerical Methods

    By     David Y. Gao, Ray W. Ogden and G.E. Stavroulakis
    Kluwer Academic Publishers, Dordrecht/Boston
        Hardbound, ISBN 0-7923-6786-3
        February 2001, 516 pp.
        EUR 145.00 / USD 157.00 / GBP 99.00

The First International Symposium on Nonsmooth/Nonconvex Mechanics was held at  Virginia Tech, June 27-30, 1999.  The second one will be held in Greece, 2002, Click here for details

Chaotic Systems

Reason for chaos: Nonconvexity (multi-potential wells).
Examples: Duffing equation

- u’’ + ( ½ u² - a) u = f

The total potential of this equation is a double-well function W(u) = ò ½ (½ u₂ - a)² dt. Therefore, the equation may have multiple (at most three) solutions at each time t . Both local min and local max depend on the input f(t) and the Newton force u’’ . The two potential well produce the so-call “stranger attractors”.

_{This can be seen from the following pre-buckled large deformation nonlinear beam vibratition (see here for details)}

Chaotic bifurcation diagram

A three-dimensional view

Gyroscope Landau-Ginzburg Equation Sine-Gorden Equation Duffing equation Extended beam Model (Gao, 1996)

New phenomena in chaotic systems
         1. Meta-chaos:  Fig 1. Fig. 2 (Chaos vase)
  2. Trio-bifurcantion and trio-chaos: Life of a nonlinear beam vibration

Fig. 1,  Life of buckling beam	Fig. 2. First meta-buckling
Fig. 3. Second meta-buckling	Fig. 4. The third meta-buckling

Some movies about finite element simulations for dynamical post-buckling analysis of a rubber diaphragm created by my Ph.D. student Axinte Ionita. This is a 3-D nonconvex, nonsmooth, nonconservative dynamical system governed by large deformation hyper-elastic constitutive law and subjected to follower force.

Movie 1(5.2MB)	Movie 2(2.32MB)
Movie 3(8.9MB)	Movie 4(3.26MB)

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