SLIDE 1
Challenges in Modeling Polycrystalline Materials
Variational Problems in spaces of measures
Shlomo Ta’asan Carnegie Mellon University
SLIDE 2 Outline
- Some issues in materials modeling
- Proposed framework – variational problems in spaces of measures
- Optimization problems for special parameterized measures
Canonical example General Theory – existence results
- Homogenization problems
- Variational Evolution Equations for special parameterized measures
SLIDE 3 Defects
Points, lines, surfaces
https://goo.gl/images/arqtiu
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SLIDE 4 Modeling using measures
Examples of measures in materials description: pairwise interatomic displacements grain size distribution grain boundary character (GBCD) lattice orientation distribution … Want a measure to describe microscopic properties at each macroscopic point → Young measures
GBCD: (Rohrer)
SLIDE 5 DiPerna Measure-Valued Solutions
Generalized Young measures Describe oscillations and concentration
- The moments of the measure satisfy the PDE in the sense of distributions
- Strong uniqueness property: if strong solution exists it should coincide with it
- Application to Euler and Navier-Stokes
We will use a different concept of measure-valued solutions
SLIDE 6
Preparation
SLIDE 7 Preparation
We design the setup to deal with problems of the form
We are also interested in gradient flows.
SLIDE 8 Preparation
We design the setup to deal with problems of the form Some spaces,
Young measures
We are also interested in gradient flows.
SLIDE 9 Preparation
We design the setup to deal with problems of the form Some spaces,
Young measures
We are also interested in gradient flows.
SLIDE 10 Preparation
We design the setup to deal with problems of the form Some spaces,
Young measures
We are also interested in gradient flows.
SLIDE 11 Preparation
We design the setup to deal with problems of the form Some spaces,
Is well defined for Young measures
We are also interested in gradient flows.
SLIDE 12 Theorem 1 Proof: using duality arguments
The problem:
SLIDE 13 Our Framework
Instead of modeling with functions in a Sobolev space: Use parameterized measures
For presentation purposes we omit the treatment of concentration!
SLIDE 14 Our Framework
Instead of modeling with functions in a Sobolev space: Use parameterized measures Probability measure for each point in
For presentation purposes we omit the treatment of concentration!
SLIDE 15 Our Framework
Instead of modeling with functions in a Sobolev space: Use parameterized measures Probability measure for each point in Recovering function values
For presentation purposes we omit the treatment of concentration!
SLIDE 16 Our Framework
Instead of modeling with functions in a Sobolev space: Use parameterized measures Compatibility condition Probability measure for each point in Recovering function values
Not the usual Young measures
For presentation purposes we omit the treatment of concentration!
SLIDE 17
An Example – formal calculations
SLIDE 18
An Example – formal calculations
Classical problem
SLIDE 19
An Example – formal calculations
Classical problem Translation: The variational problem:
SLIDE 20
An Example – formal calculations
Classical problem Translation: Compatibility condition Weak formulation of the compatibility condition The variational problem:
SLIDE 21
Summarizing,
SLIDE 22
Summarizing, Using Duality (Lagrange multipliers)
SLIDE 23 Summarizing, Using Duality (Lagrange multipliers)
Giving,
Note that
SLIDE 24
The general problem
SLIDE 25 The general problem
Formal duality calculation gives,
SLIDE 26 The general problem
For convex integrand we have a unique solution and the measures can be associated with a function.
Formal duality calculation gives, Where solves the dual optimization problem
SLIDE 27
Variational Problems for Special Young Measures
SLIDE 28 Variational Problems for Special Young Measures
Study the problem
Existence? Uniqueness?
SLIDE 29
The Dual Problem
SLIDE 30
The Dual Problem
The conjugate function.
SLIDE 31
The Dual Problem
The conjugate function.
SLIDE 32 The Dual Problem
The conjugate function.
Theorem 2:
SLIDE 33 Theorem 3: Proof: using duality arguments
SLIDE 34
Outline of proof
SLIDE 35 Outline of proof
Define dual function
SLIDE 36 Outline of proof
Define dual function Linear terms in imply
SLIDE 37 Outline of proof
Define dual function Linear terms in imply
SLIDE 38
SLIDE 39
We have, Let
SLIDE 40
We have, Let
SLIDE 41
We have, Let
SLIDE 42
Summarizing,
SLIDE 43 Summarizing,
Taking a minimizing sequence and using weak compactness of measures gives existence. For the statement about the support of the measure:
SLIDE 44 Summarizing,
Taking a minimizing sequence and using weak compactness of measures gives existence. For the statement about the support of the measure: using a selection theorem (Ekland Temam Ch 8, Thm 1.2 ) we can find a measurable selection
SLIDE 45 Summarizing,
Taking a minimizing sequence and using weak compactness of measures gives existence. For the statement about the support of the measure: using a selection theorem (Ekland Temam Ch 8, Thm 1.2 ) we can find a measurable selection The measure attains the lower bound. In addition, if the support does not satisfy the condition mentioned, then it is not optimal.
SLIDE 46
A Homogenization Example
Oscillating coefficients
SLIDE 47
A Homogenization Example
Oscillating coefficients Parameterized measure
SLIDE 48 A Homogenization Example
Oscillating coefficients Parameterized measure
Assumption about the oscillations in C
SLIDE 49
The dual problem is Which gives the known result, This gives in addition to the effective equation for the weak limit, also the characterization of the oscillations, and allow calculation of all moments. where
SLIDE 50 Variational Evolution Equations
The evolution of the parameterized measure
SLIDE 51 Variational Evolution Equations
The evolution of the parameterized measure Compatibility condition implies,
SLIDE 52 Variational Evolution Equations
A potential formulation of a gradient flow, The evolution of the parameterized measure Compatibility condition implies,
SLIDE 53 Variational Evolution Equations
A potential formulation of a gradient flow, The evolution of the parameterized measure Compatibility condition implies,
This formulation does NOT reduce back to Sobolev space solution if it exists.
SLIDE 54 Motivated by minimizing movements, to derive the gradient flow for that case, and noticing that the L2 norm above is the Wasserstein distance but only in the variable, we arrive at the following gradient flow formulation
SLIDE 55 Equations in Weak Form??
Review the minimization problem, and the associated equation in weak form,
Start with And consider perturbations that preserve the total mass and the compatibility condition we arrive at, For V:
SLIDE 56 The problem: Find a special Young measure satisfying The last statement says that the support of is where Is this enough to determine the solution? This implies ?
Needs a Lax-Milgram type of theorem in Banach spaces
SLIDE 57
Thank you!
