NUMERICAL METHODS IN APPLIED STRUCTURAL MECHANICS Lecture notes: Prof. Maurício V. Donadon
Non-linear static problems
Introduction
Sources of nonlinearities in structural analysis Geometrical non-linearity Non-linear material behaviour Non-linear boundary conditions
Geometrical nonlinearities Normal strain-displacement relationships
Geometrical nonlinearities Shear strain-displacement relationships
Geometrical nonlinearities
Nonlinear material behaviour ELASTIC ELASTO-PLASTIC MICROCRACKING ELASTIC +
Nonlinear material behaviour
Nonlinear material behaviour
Nonlinear boundary conditions Transient boundary problems: Boundary conditions change during the analysis!!!
Nonlinear boundary conditions
Solution methods for non-linear static problems
Solution methods Incremental solutions Iterative solutions Combined incremental/iterative solutions Arc-length method Quasi-static solutions
General form for a static problem Example: Linear/Nonlinear Spring K = K0 f(x) Fe
Example: Linear/Nonlinear Spring
General form for a static problem Example: Linear/Nonlinear Spring Trivial solution: Displacement control Non-trivial solution: Load control which is commonly used in structural analyses!!!
Incremental solution
INCREMENTAL SOLUTION METHOD BASED ON THE EULER METHOD
THE EULER METHOD ALGORITHM
Example: Nonlinear Spring
Example: Nonlinear Spring
Example: Nonlinear Spring
Iterative solution
ITERATIVE SOLUTION BASED ON THE NEWTON RAPHSON METHOD
THE NEWTON RAPHSON ALGORITHM
Example: Nonlinear Spring
Example: Nonlinear Spring
Example: Nonlinear Spring
Combined incremental/iterative solutions
COMBINED INCREMENTAL/ITERATIVE SOLUTIONS
INCREMENTAL/ITERATIVE SOLUTION ALGORITHM
Example: Nonlinear Spring
Example: Nonlinear Spring
Example: Nonlinear Spring
Arc-length Method
Highly non-linear structural responses snap-through snap-back
Arc-length method: Constraint equation ψ: scale factor, scale forces to the same order of magnitude of the displacements qef: Fixed external load level vector
Arc-length method: Residual force Equations to be solved simultaneously
Arc-length method Subscript “o” refers to old (previous) iteration Subscript “n” refers to current iteration
Equations to be solved simultaneously Augmented stiffness matrix Arc-length method Equations to be solved simultaneously Augmented stiffness matrix
Arc-length method Error function computation:
Spherical Arc-length method The computational cost associated with the inversion of the augmented stiffness matrix during the iterations is very high because the augmented stiffness matrix is neither symmetric nor banded! Better solution: Spherical Arc-length!!!!
Spherical Arc-length method Instead of solving both constraint and equilibrium equations simultaneously, one may assume displacement control at single point. Thus, by assuming displacement control the residual forces can be written as follow,
Spherical Arc-length method δpt must be computed for the initial predictor step based on the Forward-Euler tangential predictor and it is fixed because Kt does not change during the iterations
Forward-Euler Tangential Predictor Spherical Arc-length method Forward-Euler Tangential Predictor Substituting into the constraint equation we obtain,
Spherical Arc-length method
Spherical Arc-length method δλ can be found by substituting the previous equation into the constraint equation, which leads to the following quadratic equation,
Spherical Arc-length method with, Which can be solved for δλ.The choice of the root will be based on the cylindrical Arc-length method
Spherical Arc-length method The great advantage of the spherical arc-length over the standard arc-length method is that the former only requires the factorisation of the banded symmetric tangent stiffness matrix. Therefore it avoids the use of the augmented stiffness matrix, which is neither symmetric nor banded!!!
Cylindrical Arc-length method The cylindrical Arc-length consists of setting to zero the scaling factor ψ. In practice, this parameter is not known a priori. Moreover, setting to zero the scaling factor simplifies the choice of the appropriated root of the quadratic equation used to compute δλ, that is,
Cylindrical Arc-length method Choosing the root Solution 1 Solution 2
Cylindrical Arc-length method Choosing the root
Cylindrical Arc-length method Choosing the root Having computed δλ, update displacement and load parameter
Convergence criterion Cylindrical Arc-length method Convergence criterion
Quasi-static solutions
Example: Nonlinear Spring DYNAMIC RELAXATION Example: Nonlinear Spring K = K0 f(x) M Fe(t) C
DYNAMIC RELAXATION Central difference method Critical time step computation
DYNAMIC RELAXATION Central difference method Displacement field Velocity field
DYNAMIC RELAXATION Damping definition Critical damping Rayleigh damping
EXPLICIT TIME INTEGRATION ALGORITHM Initial conditions, v0, σ0, n=0, t=0, compute M Compute acceleration an = M-1Fe,n Update nodal velocities: vn+1/2 = vn+1/2-α + αΔtan α = 1/2 if n=0 α = 1 if n>0 Update nodal displacements: un+1 = un+ Δtvn+1/2 Compute strains Compute stresses Compute internal forces Compute residual force vector: Fi - Fe Update counter and time: n = n+1, t = t+Δt If simulation not complete go to step 2
Example: Nonlinear Spring
Example: Nonlinear Spring
Example: Nonlinear Spring – 1.0 N/s
Example: Nonlinear Spring – 0.1 N/s
Example: Nonlinear Spring – 0.01 N/s
Example: Nonlinear Spring – Damping effect Thigh=0.12 Tlow=0.22
Example: Nonlinear Spring – Damping effect
Example: Nonlinear Spring – Damping effect
Example: Nonlinear Spring – Damping effect
Example: Nonlinear Spring – Damping effect
Over damping effects in dynamic relaxation Over damping MUST BE AVOIDED in dynamic relaxation methods! Special care must be taken with over damping Over damping increases artificially the internal energy of the system!!!!