Variational methods for damage evolution Tuesday, 10/9/2007, 3:30 PM-4:30 PM
SPEAKER: Christopher J. Larsen (WPI)
ABSTRACT: Variational methods have been very successful for static problems in materials science: they have explained microstructure in martensite, predicted regions of damage in elastic materials, and produced displacements with cracks that are stable in the sense of Griffith's criterion, to name a few. Recently (in the past ten years or so), there have been attempts to extend these methods to evolution problems, specifically, to quasi-static problems, with the hope of ultimately generating well-posed dynamic models -- currently, there are no reasonable models for crack dynamics or damage evolution with dynamics. In this seminar, I will describe current quasi-static models for fracture and damage together with some mathematical issues in the existence proofs.
Finally, I will discuss what I view as some deficiencies, and why there is some hope of getting at dynamics.
Completeness, and Liouville Property of a Non-Compact Weighted Manifold Tuesday, 10/23/2007, 3:00 PM-4:00 PM
SPEAKER: Jun Masamune (WPI)
ABSTRACT: A weighted manifold $M$ is a manifold furnished with a Riemann tensor and a measure which has a smooth density against the Riemann measure. It carries a second-order elliptic operator called the weighted Laplacian. A weighted manifold $M$ is said to be stochastic complete if the Brownian motion associated to the weighted Laplacian can be found in $M$ for any positive time. In this talk we will discuss a Liouville type property which implies the stochastic completeness and observe that the stochastic completeness implies the essential self adjointness of the weighted Laplacian of a non-compact weighted manifold.
We will also observe that if the Cauchy boundary of $M$ is almost polar, then the weighted Laplacian is essentially self-adjoint.
Variational Methods for Damage Evolution, Part II Tuesday, 10/30/2007, 3:00 PM-4:00 PM
SPEAKER: Christopher Larsen
ABSTRACT: Variational methods have been very successful for static problems in materials science: they have explained microstructure in martensite, predicted regions of damage in elastic materials, and produced displacements with cracks that are stable in the sense of Griffith's criterion, to name a few. Recently (in the past ten years or so), there have been attempts to extend these methods to evolution problems, specifically, to quasi-static problems, with the hope of ultimately generating well-posed dynamic models -- currently, there are no reasonable models for crack dynamics or damage evolution with dynamics. In this seminar, I will describe current quasi-static models for fracture and damage together with some mathematical issues in the existence proofs.
Finally, I will discuss what I view as some deficiencies, and why there is some hope of getting at dynamics.
On lubrication equations in thin liquid films Tuesday, 11/6/2007, 3:00 PM-4:00 PM
SPEAKER: Burt S. Tilley (WPI and Olin College)
ABSTRACT: Interfacial fluid dynamics centers on understanding the evolution and pattern formation of a sharp interface between two fluids. The evolution of this interface depends on the flow local to the interface, and this flow, in turn, depends on the interface shape and motion. A denouement can be achieved in situations where the aspect ratio of the fluid problem is small. Through a regular
perturbation expansion in powers of the aspect ratio, effective equations of motion can be derived that describe the evolution and dynamics of the interface, and identifies physical mechanisms of instability. We consider the dynamics and patterns that arise from these equations for spatially periodic solutions and for solutions over bounded spatial domains. Examples, as time permits, come from problems in coating problems, heat-exchange devices, inkjet printing applications, and respiratory treatments in the human airway.
PDE Seminar: "Analysis on Some Parabolic Systems (strongly coupled or higher order)" Tuesday, 12/4/2007, 3:00 PM-4:00 PM
SPEAKER: Li CHEN (Department of Mathematical Sciences, Tsinghua University & Harvard University)
ABSTRACT: In this talk, I will give a brief outline of the work we have done on some strongly coupled parabolic systems and fourth order parabolic equations, on which classical techniques such as comparison principles for parabolic equations could not work. The models are from diffusion systems in semiconductor simulation and population models in biomathematical models. Our ideas are mainly based on the exponential transformation and entropy inequalities. We will give the global existences and large time behavior of the weak solutions.
PDE Seminar "Fracture Paths from Front Kinetics: Relaxation and Rate-Independence" Tuesday, 1/15/2008, 3:00 PM-4:00 PM
SPEAKER: Casey Richardson (WPI)
ABSTRACT: Crack fronts play a fundamental role in engineering models for fracture: they are the location of both crack growth and the energy dissipation due to growth. However, there has not been a rigorous mathematical definition of crack front, nor rigorous mathematical analysis predicting fracture paths using these fronts as the
location of growth and dissipation. In this talk, I will discuss my joint work with Chris Larsen and Michael Ortiz on front based fracture models. I will present a natural weak definition of crack front and front speed, and discuss the analysis of modelsof crack growth in which the energy dissipation is a function of the front speed, i.e., the dissipation rate at time $t$ is of the form \[\int_{F(t)} \psi(v(x,t)) d\mathcal{H}^{N-2}(x)\] where $F(t)$ is the front at time $t$ and $v$ is the front speed. Then I will discuss our main result: an $N$-dimensional relaxation formula that gives the surprising result that the effective dissipation is rate-independent for \emph{any choice of $\psi$}.
PDE Seminar - Homogenization of a Problem on a Domain with Oscillating Boundaries via Periodic Unfolding Wednesday, 4/9/2008, 3:00 PM-4:00 PM
SPEAKER: Alain Damlamian
ABSTRACT: A variational problem on a sequence of 2-dimensional domains with oscillating boundaries is studied. Using the periodic unfolding method, the homogenized problem is obtained in the limit as the period length approaches zero. Several extensions are also given. In this framework, a result of strong convergence is obtained which is new.
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Variational methods for damage evolution Tuesday, 10/9/2007, 3:30 PM-4:30 PM
SPEAKER: Christopher J. Larsen (WPI) ABSTRACT: Variational methods have been very successful for static problems in materials science: they have explained microstructure in martensite, predicted regions of damage in elastic materials, and produced displacements with cracks that are stable in the sense of Griffith's criterion, to name a few. Recently (in the past ten years or so), there have been attempts to extend these methods to evolution problems, specifically, to quasi-static problems, with the hope of ultimately generating well-posed dynamic models -- currently, there are no reasonable models for crack dynamics or damage evolution with dynamics. In this seminar, I will describe current quasi-static models for fracture and damage together with some mathematical issues in the existence proofs. Finally, I will discuss what I view as some deficiencies, and why there is some hope of getting at dynamics.
Completeness, and Liouville Property of a Non-Compact Weighted Manifold Tuesday, 10/23/2007, 3:00 PM-4:00 PM
SPEAKER: Jun Masamune (WPI) ABSTRACT: A weighted manifold $M$ is a manifold furnished with a Riemann tensor and a measure which has a smooth density against the Riemann measure. It carries a second-order elliptic operator called the weighted Laplacian. A weighted manifold $M$ is said to be stochastic complete if the Brownian motion associated to the weighted Laplacian can be found in $M$ for any positive time. In this talk we will discuss a Liouville type property which implies the stochastic completeness and observe that the stochastic completeness implies the essential self adjointness of the weighted Laplacian of a non-compact weighted manifold. We will also observe that if the Cauchy boundary of $M$ is almost polar, then the weighted Laplacian is essentially self-adjoint.
Variational Methods for Damage Evolution, Part II Tuesday, 10/30/2007, 3:00 PM-4:00 PM
SPEAKER: Christopher Larsen ABSTRACT: Variational methods have been very successful for static problems in materials science: they have explained microstructure in martensite, predicted regions of damage in elastic materials, and produced displacements with cracks that are stable in the sense of Griffith's criterion, to name a few. Recently (in the past ten years or so), there have been attempts to extend these methods to evolution problems, specifically, to quasi-static problems, with the hope of ultimately generating well-posed dynamic models -- currently, there are no reasonable models for crack dynamics or damage evolution with dynamics. In this seminar, I will describe current quasi-static models for fracture and damage together with some mathematical issues in the existence proofs. Finally, I will discuss what I view as some deficiencies, and why there is some hope of getting at dynamics.
On lubrication equations in thin liquid films Tuesday, 11/6/2007, 3:00 PM-4:00 PM
SPEAKER: Burt S. Tilley (WPI and Olin College) ABSTRACT: Interfacial fluid dynamics centers on understanding the evolution and pattern formation of a sharp interface between two fluids. The evolution of this interface depends on the flow local to the interface, and this flow, in turn, depends on the interface shape and motion. A denouement can be achieved in situations where the aspect ratio of the fluid problem is small. Through a regular perturbation expansion in powers of the aspect ratio, effective equations of motion can be derived that describe the evolution and dynamics of the interface, and identifies physical mechanisms of instability. We consider the dynamics and patterns that arise from these equations for spatially periodic solutions and for solutions over bounded spatial domains. Examples, as time permits, come from problems in coating problems, heat-exchange devices, inkjet printing applications, and respiratory treatments in the human airway.
PDE Seminar: "Analysis on Some Parabolic Systems (strongly coupled or higher order)" Tuesday, 12/4/2007, 3:00 PM-4:00 PM
SPEAKER: Li CHEN (Department of Mathematical Sciences, Tsinghua University & Harvard University) ABSTRACT: In this talk, I will give a brief outline of the work we have done on some strongly coupled parabolic systems and fourth order parabolic equations, on which classical techniques such as comparison principles for parabolic equations could not work. The models are from diffusion systems in semiconductor simulation and population models in biomathematical models. Our ideas are mainly based on the exponential transformation and entropy inequalities. We will give the global existences and large time behavior of the weak solutions.
PDE Seminar "Fracture Paths from Front Kinetics: Relaxation and Rate-Independence" Tuesday, 1/15/2008, 3:00 PM-4:00 PM
SPEAKER: Casey Richardson (WPI) ABSTRACT: Crack fronts play a fundamental role in engineering models for fracture: they are the location of both crack growth and the energy dissipation due to growth. However, there has not been a rigorous mathematical definition of crack front, nor rigorous mathematical analysis predicting fracture paths using these fronts as the location of growth and dissipation. In this talk, I will discuss my joint work with Chris Larsen and Michael Ortiz on front based fracture models. I will present a natural weak definition of crack front and front speed, and discuss the analysis of modelsof crack growth in which the energy dissipation is a function of the front speed, i.e., the dissipation rate at time $t$ is of the form \[\int_{F(t)} \psi(v(x,t)) d\mathcal{H}^{N-2}(x)\] where $F(t)$ is the front at time $t$ and $v$ is the front speed. Then I will discuss our main result: an $N$-dimensional relaxation formula that gives the surprising result that the effective dissipation is rate-independent for \emph{any choice of $\psi$}.
PDE Seminar - Homogenization of a Problem on a Domain with Oscillating Boundaries via Periodic Unfolding Wednesday, 4/9/2008, 3:00 PM-4:00 PM
SPEAKER: Alain Damlamian ABSTRACT: A variational problem on a sequence of 2-dimensional domains with oscillating boundaries is studied. Using the periodic unfolding method, the homogenized problem is obtained in the limit as the period length approaches zero. Several extensions are also given. In this framework, a result of strong convergence is obtained which is new.
Powered by the Social Web - Bringing people together through Events, Places, & Common Interests