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**LIQUEFACTION, CRITICALITY AND COMPLEX SYSTEM**

Jeen-Shang Lin Department of Civil and Environmental Engineering University of Pittsburgh 4/30/09 NTOU

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**Kobe-Pitt Symposium on Disaster Risk Reduction and Response**

Point Based Criteria qc, (Mpa) fs, (kPa) stress resistance Depth (m) 9/6/2007 Kobe-Pitt Symposium on Disaster Risk Reduction and Response

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**Point failure criteria**

Motivation Identification of sites that are susceptible to liquefaction triggered by earthquakes is a crucial part of an earthquake hazard mitigation effort. Point failure criteria

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**Questions to be answered:**

Is the soil susceptible to liquefaction? If the soil is susceptible, will liquefaction be triggered? If liquefaction is triggered, will damage occur? Answers to these questions require tackling some deep questions…..

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**Common large scale ground failures:**

Liquefaction Landslide Ground movement (uplift or subsidence) Debris flow Mud flow 9/6/2007 Kobe-Pitt Symposium on Disaster Risk Reduction and Response

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Credit: C-Y KU

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**What is the deeper question?**

Reflections on Failure Failures often have common fundamental physics and structures: Intrinsic nature, invariant scale measure. It is important to view them from a unified framework. Numerical analysis is difficult but not insurmountable. We are very good at computation. But shouldn’t a computation be guided by a proper understanding of physics? 9/6/2007 Kobe-Pitt Symposium on Disaster Risk Reduction and Response

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**Intrinsic failure: Topology of a problem?**

Key block theory + discrete elements

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**Stresses induce fracture and block movements**

47 removable out of 57 Simplified modeling of flow stress 53 removable out of 63

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**Three ways a soil may respond depending on their states (Intrinsic nature?)**

Castro (1969) liquefaction failure dilative response limited liquefaction failure Monotonic undrained triaxial tests on Banding Sands

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**Conventional computational approaches**

Constitutive modeling + FEM work quite well, but does not address a more fundamental question.

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**Evolution of Slope Failure**

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**Mesh Based Partition of Unity: A new paradigm**

Problem domain Mathematical Mesh This numerical method is based upon manifold method generalize the construct using the partition of unity the framework. Later on in this presentation we introduce two dimensional key block into the analysis. The manifold method is rooted in the discrete element method and has been used in solving discrete continuum problems. It basically allows the flexibility of giving a discrete block any degrees of freedom. So one can have a mixture of blocks of various degrees of freedom. We have a mathematical mesh to provide the nodes for building the finite covering of the solution domain. Physical Mesh 13

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**Same foundation with meshless methods: Partition of unity**

Diffusion element Element free Galerkin method Reproducing Kernel Particle Method h-p cloud method Partition of unity finite element method Extended finite element method Generalized finite element method All these methods has the same partition of unity in their construct. The functions may be derived differently. The most common way of deriving the partition of unity function is through moving least square. (Do you need more…) 14

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**Covers, nodes and Elements**

In the case of triangular mathematical mesh, a local cover Ѡα, centered at node α, is formed by the union of all triangular areas that share the common node. A local cover is an open set whose closure is a neighborhood star. Using the finite element analogy, cover are nodes and overlapped covered area are elements. For this triangular i-j-k, interior of the element is covered by three overlapped local covers. An element in the this method is an area covered by a fixed number of covers. Covers are nodes Overlapped covers are elements 15

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**The use of finite covers leads to node based computation**

Element Free Galerkin method Nodes lie inside a problem domain. Boundary, covers and supports (Belytschko et al ) Meshed Based Partition of Unity No restriction on nodes locations (XFEM) There may be differences in the use of finite covers. For instance, in EFG, the nodes lie within a problem domain. Whereas in the present method, and in Extended FEM, nodes can lie outside the problem domain. Any nodes that have cover functions go outside the problem domain requires care in integration. 16

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**Formulation Partition of Unity Function: Duarte and Oden (1996)**

Approximate Function In the original manifold method, Shi used term “weighting function” for “the partition of unity function”. This φα is weighting function in manifold. Duarte and Oden have shown further that partition of unity can be negative. The manifold approximation on a cover defined at a node α, can be expressed as product of a polynomial basis p(x), with the partition of unity. For instance for a constant strain FEM, the polynomial degenerated into a constant, while the cover function is a simple bilinear function. Discrete equation can be obtain from Galerkin formulation. Weak form of the discrete problem with penalty formulation 17

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**Discontinuity Modeling: Add nodes**

In the case of discontinuity present, manifold add cover so that there is one cover associated with each partition. In other words, one can have two nodes for a cover that is cut by a discontinuity into two partitions. Each cover is defined with an identical partition of unity function with a modifier. 18

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**Mesh Based Partition of Unity: A new paradigm**

Failure computation has made substantial Pr Mesh Based Partition of Unity: A new paradigm Problem domain Mathematical Mesh This numerical method is based upon manifold method generalize the construct using the partition of unity the framework. Later on in this presentation we introduce two dimensional key block into the analysis. The manifold method is rooted in the discrete element method and has been used in solving discrete continuum problems. It basically allows the flexibility of giving a discrete block any degrees of freedom. So one can have a mixture of blocks of various degrees of freedom. We have a mathematical mesh to provide the nodes for building the finite covering of the solution domain. Physical Mesh 19

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**Same foundation with meshless methods: Partition of unity**

Diffusion element Element free Galerkin method Reproducing Kernel Particle Method h-p cloud method Partition of unity finite element method Extended finite element method Generalized finite element method All these methods has the same partition of unity in their construct. The functions may be derived differently. The most common way of deriving the partition of unity function is through moving least square. (Do you need more…) 20

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**Covers, nodes and Elements**

In the case of triangular mathematical mesh, a local cover Ѡα, centered at node α, is formed by the union of all triangular areas that share the common node. A local cover is an open set whose closure is a neighborhood star. Using the finite element analogy, cover are nodes and overlapped covered area are elements. For this triangular i-j-k, interior of the element is covered by three overlapped local covers. An element in the this method is an area covered by a fixed number of covers. Covers are nodes Overlapped covers are elements 21

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**The use of finite covers leads to node based computation**

Element Free Galerkin method Nodes lie inside a problem domain. Boundary, covers and supports (Belytschko et al ) Meshed Based Partition of Unity No restriction on nodes locations (XFEM) There may be differences in the use of finite covers. For instance, in EFG, the nodes lie within a problem domain. Whereas in the present method, and in Extended FEM, nodes can lie outside the problem domain. Any nodes that have cover functions go outside the problem domain requires care in integration. 22

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**Formulation Partition of Unity Function: Duarte and Oden (1996)**

Approximate Function In the original manifold method, Shi used term “weighting function” for “the partition of unity function”. This φα is weighting function in manifold. Duarte and Oden have shown further that partition of unity can be negative. The manifold approximation on a cover defined at a node α, can be expressed as product of a polynomial basis p(x), with the partition of unity. For instance for a constant strain FEM, the polynomial degenerated into a constant, while the cover function is a simple bilinear function. Discrete equation can be obtain from Galerkin formulation. Weak form of the discrete problem with penalty formulation 23

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**Discontinuity Modeling: Add nodes**

In the case of discontinuity present, manifold add cover so that there is one cover associated with each partition. In other words, one can have two nodes for a cover that is cut by a discontinuity into two partitions. Each cover is defined with an identical partition of unity function with a modifier. 24

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**Back to Science Liquefaction : What happened?**

Casagrande had envisioned that a “flow structure” would develop at liquefaction (Castro, 1969): “….during a liquefaction slide the relative position of the grains is constantly changing in a manner which contains a minimum resistance. He explained that the change from a normal structural arrangement of the grains to be flow structure would start almost accidentally in a nucleus and then spread through the mass by a chain reaction, and that such a reaction could explain the spontaneous character of liquefaction slide.”

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**Liquefaction as a Critical Behavior of a Complex System?**

A sand assembly is a complex system, and “critical state”, “phase transition”, “steady state” are also the focus points in the dynamics of various complex systems, including even biological systems (e.g., Kauffman, 1993). On the outset, a steady state in the terminology of nonlinear dynamics is often referred to as a “fixed point” or a “fixed point attractor” (e.g., Williams, 1997), as a system is attracted to a fixed state under sustained loading. The physics governing the process that leads to an attractor generally have something in common. It is believed here that linking liquefaction to the current understanding of nonlinear dynamics may also provide a better understanding for post-liquefaction behavior.

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**Self-Organized Criticality**

Castro and Poulos (1977) have proposed an interpretation for cyclic mobility. They postulated that large strains resulting from cyclic mobility in laboratory on dilative sands are due principally to redistribution of void ratio within the specimen during cyclic loading. In 1987, Bak, Tang and Wisenfeld proposed the concept of “self-organized criticality” (SOC). They discovered that large, dissipative complex system had the tendency of driving themselves into a critical state with wide range of length and time scale.

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**Kobe-Pitt Symposium on Disaster Risk Reduction and Response**

A percolation view? There can’t be no flow unless voids are connected into a path through a material domain. 9/6/2007 Kobe-Pitt Symposium on Disaster Risk Reduction and Response

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**Failure of a uniform material**

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**Failure of a heterogeneous material**

A global failure requires local failure to be connected….

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**Failure of a heterogeneous material**

Failure is not static

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Critical Threshold How wide spread needs the liquefaction be within a given site in order for it to fail? We used percolation theory to address this issue: A problem domain is first discretized with a regular lattice Each lattice cell may be vacant or occupied with an occupancy rate, p. We then asked, “What is occupancy level in a lattice at which it is possible to find a pathway through the unoccupied population of the lattice?” The critical occupancy rate is called a percolation threshold, or critical threshold, pc. One of our central issues is: “What portion of a site has to be liquefied before a site is considered liquefied?”

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**Renormalization Group Method**

Basic Failure group for Von Mises materials pc is found to be 0.618

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Critical threshold pc is found to be 0.618

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**Verification of Percolation failure**

We conduct strain control compression test simulation on Von Mises material using samples consists of 10,000 cells. Two strength levels were selected; each cell was assigned one of the two strengths according to the random number generated. If the random number, which lay between 0 and 1, selected was less or equal to an assigned p, the lower strength was given; otherwise the higher strength was used.

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Take a detailed look Cluster size distribution

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**Uniaxial experiment: bonded particles (initial void ratio=0.16)**

Red: location of failed bonds Deviatoric stress vs Axial strain

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**Failure Progression Breaking of bonds**

0.066% axial strain 0.165% 0.198% 2.31%

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**same sample L=H (13203 particles)**

Deviatoric stress vs Axial strain

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**Failure Progression Breaking of bonds**

0.198% axial strain 0.264%

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**Using an assembly of particles --Looking from a different angle**

using an assembly of particles and conduct numerical experiment for understanding

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**A sample triaxial test include unloading reloading**

Devistoric stress vs axial strain WE are in the process of tracing the failure location and cluster size, as well as what happen at the largestrain. Volumetric strain vs axial strain

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Concluding remarks This is a work in progress. We are now using both particle codes and continuum code with bounding surface model with state variables to study the problem involving self –organization after failure. We believe that looking into a scale invariance measure, the phase change from percolation view point can be invaluable.

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**Cell Traction Force Microscopy Method**

Jeen-Shang Lin In Collaboration with James H-C. Wang MechanoBiology Laboratory Departments of Orthopaedic Surgery University of Pittsburgh

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Cell traction force microscopy (CTFM) is recognized as a simple and effective method for quantifying cell traction forces. Migration Traction forces polyacrylamide gel Substrate Cell

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A B “Force-loaded” image “Null-force” image

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**Image Registration Feature-based registration**

Non-Feature Based Registration

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Phase congruence In the frequency domain, at locations of complex image intensity variation, the local Fourier components are maximally in phase. local phase is also invariant to image brightness or contrast. To determine the phase information spatially, one needs to process the images to simultaneously obtain the spatial and the frequency information. Use filter bank: Gabor functions

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**Feature identification: A Phase Congruence Method**

B C

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**Selected Features in null force image**

Matched featured in force loaded image

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**A differential based registration**

Models a mapping function for a given pixel (x,y) from the target Io to source I1 in the following form m be locally smooth

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A A B

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**Vulnerability Assessment of Critical Infrastructure**

Jeen-Shang Lin Department of Civil and Environmental Engineering University of Pittsburgh

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**The deficiency of old mode of thinking**

Designing a functional system is not enough. Form is as important as function. Topology robustness. We need a new set of tools and metrics. Shifting from trees to woods mindset Easy to read a tree, but how do you read the woods?

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**Conventional Risk Assessment Study**

Identify failure mode of a component Develop a fragility of a component Identify Hazard Develop of a hazard map with various return periods Integration and obtain the risk

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Small World Network A small world is a world of order that embodies a certain degree of disorder. (The world is not ordered, nor is it completely random.) It originated from the study of social network: It is believed that almost any pair of people in the world can be connected to one another by a short chain of intermediate acquaintances. Stanley Milgram (1967) the chain is of order six thus six degree of separation. Watts and Strogatz, 1998, the small world model The small world effect appears to be universal and applies to other networks.

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**Characteristics of a Small world network**

Sparse: A network has relatively few links. An n-node network of links much less than Clustered: It is not random Links are not uniformly distributed; Small diameter: For a small world network, the largest separation is much less than n, and is closer to ln(n).

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**There are some things about a graph…**

Characteristic Path Length, L Medium of the averages of shortest path between any two nodes (2.58) Diameter: (Degree of separation) Largest shortest path between any two vertices (6) Clustering coefficient: How well are your neighbors connected? Shortcuts, contraction,….

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**Workshop on Hazard Reduction and Responses in Metropolitan Regions**

The random rewiring model Watts, 1999 Workshop on Hazard Reduction and Responses in Metropolitan Regions

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Original: k=3.29, L=4.4,D=8,c=0.5,phi=0.195;psi=0.213 After adding three links: k=3.5, L=3.72,D=8,c=0.442,phi=0.245;psi=0.331

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**Workshop on Hazard Reduction and Responses in Metropolitan Regions**

March 16-18, 2003, J-S Lin Workshop on Hazard Reduction and Responses in Metropolitan Regions

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