Material Point Method Solution Procedure Map from particles to grid Interpolate from grid to particles Constitutive model Boundary conditions Wednesday, 10/9/2002 Mass matrix F (n) +f (n) = m
(n,n') (n') a n' Compact mass matrix: Only matrix elements close to diagonal are not zero. Lumped Mass Matrix F (n) + f (n) = m(n,n')a(n') M (n)a(n) n' F (1) +f (1) m(1,1) m(1,2) F (2) +f (2) m(2,1) m(2,2) (3) (3)
m(3,2) F +f = 0 M 0 0 (n) (n) 0 F +f 0 0 m(2,3) m(3,3) 0 0 m(3,4) L
0 L m(n,n1) F (1) +f (1) M (1)a(1) F (2) +f (2) M (2)a(2) (3) (3) (3) (3) F +f =M a M M F (n) +f (n) M (n)a(n)
0 a(1) 0 a(2) 0 a(3) L M (n,n) (n) m a Solution 1. 2. 3. 4. 5. 6. Particle discretization, grid Initialize particle information Map particle information to grid
Solve the motion equations on grid Interpolate from grid to particles Update the information on particles Particle information position: x( p) velocity: v( p) stress: ( p) Mapping: particle grid External force
F (n) = m b N (p) (p) (n, p) p Internal force Mass matrix (n, p) dN
f (n) = V (p) (p) dx p M (n) = m( p)N (n, p) p Momentum M (n)v(n) = m(p)v( p)N(n, p) p Solve Equations on the Grid 1 (n) (n) a = (n) (F +f ) M (n)
v(n) v(n) +a(n) t Update particle position ( p) x ( p) x 1 ( p) (n) (n, p) + v + v N t 2 n Update particle velocity
v( p) v( p) +t a(n)N(n, p) n Update particle stress Linear elastic constitutive model ( p) ( p) +E ( p)t E is Youngs modulus. Particle strain rate ( p) d ( p) = v dx ( p) = v(n )
n d (n, p) N dx ( p) ( p) +E ( p)t = ( p) +Et v(n) n d (n, p) N dx Stress-strain curve ( p)
( p) d ( p) d ( p) (n ) d (n, p ) +t = +t v N d d n dx Boundary Conditions F
(n) +f (n) (n) (n) =M a If v(1)<0: contact Momentum change: M (1)v(1) M (1)v(1) F (1)t =(M (1)v(1)) M (1)v(1) =2M (1)v(1) F
(1) 2M (1)v(1) = t Next few classes 2D MPM How to write MPM code in MATLAB Techniques in programming, data structure
To maximize economic efficiency, including preventing or making right significant societal or consumer harm that results from market imperfections Theories of Regulation Private interest theories Peltzman (1976) - Self-interested regulators engage in regulatory activities consistent with maximizing their political support.
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