Domain

Explanation

TOC

• Dynamics:
• Fundamentals:

1. SDOF
2. MDOF
3. 1-D continuous systems

• FEA for dynamics:

1. Beams
2. 2-D solids
3. Numerical solutions of FEA

• Heat transfer analysis
• Analysis of field problems
• Viscous incompressible fluid flow
• Numerical solutions of FEA
• Analysis of coupled problems

Dynamics

• Fundamentals
• Frequency domain method
• Intentions:
• Select cost effective scheme
• Select appropriate parameter values & time sizes
• Objectives:
• Formulate & solve FE equations for dynamics
• Mode superposition:
• Good: linear, very small nonlinearity, few modes from loadings
• Loading rates: slow (fewer lower modes), fast (more lower modes)
• Bad: for nonlinear, as M,C & esp. K has to change at every time step; too time-consuming
• Direct time integration:
• Explicit: faster to operate; no inversion; lower stability & accuracy; better for linear & little non-linear
• Stability depends on time step:
• Better efficiency: no need for matrix inversion
• Derive discrete FDM scheme
• Forward difference: less stable, accumulated error
• Backward difference: more stiff (due to stiffness changes not reflected), less responses
• Central difference method (CDM): balanced, more stable, less stiff, less accumulated error, more responses
• Needs assumed or given I.C. to give unique responses
• Newmark:
• Implicit: slower to operate; inversion for simultaneous solution at each time step; better stability & accuracy
• Best for nonlinear accuracy
• Assumed: linear acceleration variations
• Gamma=0: previous point; gamma=1: next point
• Characteristics: determined by parameters gamma & beta
• Stability:
• No error accumulations
• Stable: large does not amplify lowest mode responses
• Time integration scheme:
• Stable: if spectral radius or, for multiple eigenvalues,
• i.e. y no accumulations
• Discrete illustration:
• 
• formulation:
• Structural dynamics:
• w_l: highest significant frequency in loading
• Select FE mesh to give all frequencies up to about w_f=4w_l
• For high-freq. loads: w_f=w_l (ok)
• Implicit: unconditionally stable à
• Wave propagation:
• 
• Courant-Friedrichs-Lewy (CFL condition):

Low-velocity contact-impact of bodies

• Impact: projectile/particle impacts/hits/collides/penetrates the host/target
• Low-velocity: 10-15 m/s or 35-45 km/h
• Objectives:

1. Extend FEM for low-velocity contact-impact of multi-bodies
2. Establish contact interface condition
3. Formulate non-linear FE equation using weak form incorporating contact interface condition & solution procedures

• High velocity:
• >15m/s: Lagrangian (slow) approach not suitable
• Use: meshless methods (astro-physics); part Eulerian (fast) part Lagrangian (slow)
• Contact interface condition criteria:

1. Kinematics: compatibility
2. Kinetics: force transfer; velocity changes at contact; contact impenetrability
3. Interpenetration: interaction at impenetration

• Intended outcomes:
• Contact interface formulations: e.g. page drop test; vehicle crashworthiness; ballistic missile penetration
• Contents:

1. Introduction
2. Impact of particle on body
3. Contact interface condition: 2 bodies à deformations, interactions, compatibility, continuity assumptions
4. Treatment of contact constraint: how to satisfy
5. Weak form & FE formulations: contact impact problem
6. Geometric considerations & solution schemes

• Geometric difficulties:
• Location of impacts: node-node, off the node impact (more common, more difficult)
• Forces: easier using equilibrium; normal; tangential: frictional: static, sliding (difficult)
• Surface: difficult using compatibility; curved surface; flat surface: but curved upon impact à changes in transfer (difficulty)
• R&D findings:

1. Behaviour simulated not accurate à must be verified by experiments (nuclear simulations?)
2. More contact forces: larger projectile mass, larger projectile velocity/momentum, more target deformations

Low-velocity impact of particle on body

• Assumptions:

1. Particle/projectile: no deformation
2. Constitutive condition follows Hertz or indentation law

• Weak FE form: (target virtual work) + (particle virtual work) + (impact virtual work) = 0
• Indentation law:

1. Needs experimental inputs
2. Hertz contact law
3. With Hertz contact constant: laminated, homogeneous plates

Summary

• Contact interface condition:

1. Kinematics: interpenetration
2. Kinetics
3. Contact conditions: that are required to be satisfied by both kinematics & kinetics

• Treatment of contact constraints: how to treat the contact conditions

1. Lagrange methods: Lagrange multiplier method, perturbed Lagrangian, augmented Lagranigian
2. Penalty approach: predictor (to allow overlapping penetration initially); corrector (to correct these overlapping with modeled corrections)
3. Friction force: < à stick together;

à move together & slide

• Weak form & FE formulation:

1. Contribution from contact
2. Geometry of master (target being impacted) surface: normally curved shell surface à difficult
3. Discretizaion of contact surfaces

• Contact search & solution schemes:

1. Contact searching: global search for locations of contact points à impt.: initial contact point(s) à easier: subsequent contact points: search in the neighbourhood of initial contact point(s)
2. Time integration schemes

• Conclusions

Contact-interface condition

• Kinematics:
• Penalty method: allows erroneous to occur, then apply corrective actions to remedy
• Interpenetrability measures for contact-impact of any 2 bodies:

1. Impenetrability
2. Conservation of momentum: across the interface
3. Normal traction across contact interface always in compression, not in tension

• Kuhn-Tucker complimentary form for contact to be present & satisfied:

1. : non-zero normal gap
2. : -ve normal traction always present
3. : t_N=0 for no contact; g_N=0 for contact
4. 

: CoM

Kinematics

• Impenetrability conditions:
• Rate form:
• Relative normal velocity
• Relative tangential velocity
• Gap function: (gap: scalar à distance between 2 points)
• Tangential gap g_T
• Normal gap g_N
• Allows overlapping first à formulate à correct later: using "spring" equivalent

Kinetics

• CoM across the contact interface requires that resultant tractions vanish:
• 

How to treat contact conditions

• Lagrange multiplier approach:

1. Others: perturbed Lagrangian, augmented Lagrangian
2. Advantages: no user-set parameters (theoretically can be exact; but practically may not be accurate due to unrealism) à satisfy contact constraints
3. Advantages: Good for static & low-velocity impact
4. Disadvantages: need to set up a nodal & element topology for Lagrange multiplier
5. Disadvantages: large sharp changes in velocities & accelerations (same as Han’s hysteresis) due to infinite spring at contacts for gap function
6. Disadv: large noisy oscillations for Lagrangian

• Penalty approach:

1. Assume: allow overlapping penetrations
2. Assume: impenetrability constraints not fully satisfied
3. Assume: spring of appropriate stiffness à penalty parameter
4. Penalty parameter: determined by numerical experiences
5. But for explicit schemes, large penalty à large numerical instability
6. Adv: number of unknowns unchanged
7. Adv: system of equations +def
8. Adv: relatively simple to implement

Example

• At each time interval, possibility of penetration à since assume: no inter-penetration
• Use penalty approach to correct inter-penetration to reach impenetration à needs contact location & rebound (equivalent contact spring stiffness, K_c)
• Contact location w.r.t. min. distance constraint à tangent to the segment à normal vector à contact point A

Tutorial

• Condition required to be satisfied under contact:
• Global coord of each node à position vector of contact point à local coord of contact point à min. distant constraint with min. orthogonal distance à solve for local coord location for contact point à updates of master (target) contact point(s) & impactor (projectile) location at each time step
• A: master element, segment or node
• B: impactor movement
• i: contact which points on master

Contact searching & solution scheme

• Search for contact point location as it moves or changes or (dis)appears
• 2 steps:

1. Global search for contact location
2. Set up local kinematic relations

• Contact searching:

1. Neighbourhood identification: set of slave contact nodes possibly impacting segment of possible master contact elements à if known: goto step 2 à if uncertain: use all slave points of previous time step; or good starting points – slave nodes from previous time step
2. For each slave node, N_s, locate nearest master node, N_m
3. Locate contact point on the master element: any element in the patch sharing N_m should be checked
4. Check whether N_s has penetrated master segment: if penetrate, penalty approach used to serve as pull-out force & contact rebound spring stiffness, K_c is reformulated
5. For each slave node, search is terminated when no contact possible
6. Repeat steps 1-5 for master nodes, if necessary

• Conditions to satisfy to locate whether N_s is on or penetrate the master element, E_i

Geometry of master surface

• Determine contact surface location changes & contact spring stiffness, K_c=K_G+K_N+K_F
• Contact spring = Geometry + Normal impact + Friction (static or sliding traction)
• General for all types of elements
• Contact point A depends on: movement of contact point & movements of contact surface

Conclusions

• Weak FE formulations: (master contribution) + (impactor contribution) + (external loads & impacts) = 0
• Indentation law
• Body-Body:

1. Gap function: g_N, g_T
2. Contact condition: ; ; ;

• Penalty approach:

1. Elasto-plastic analogy: similar solution for penalty approach
2. t_N, t_T
3. Allows penetration à contact spring: if infinite – too much noises à correction

• Location of contact point on contact surface: from min. distant constraint
• Contribution from contact: K_c=K_G+K_N+K_F
• Contact search procedure