8.2.3 Cables and unilateral contact analysis
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1 Operational Programme Education and Lifelong Learning Continuing Education Programme for updating Knowledge of University Graduates: Modern Development in Offshore Structures AUTh TUC Cables and unilateral contact analysis Georgios E. Stavroulakis Professor, Technical University of Crete, Department of Production Engineering and Management
2 Cable soil interaction: unilateral contact Unilateral contact (Stavroulakis, Baniotopoulos, Panagiotopoulos 1986)
3 Presentation Unilateral contact Nonsmooth effects in nature - mechanics Motivation and recent applications Non-smooth, convex analysis Variational Inequalities Non-convex analysis and optimization Hemivariational inequalities, local minima Applications Modelling, Inverse Problems
4 Presentation for FEM Users Industrial unilateral contact and friction (using MARC finite element software) More general nonsmooth interactions Notice that nonmonotone laws are provided by FEM, but without theoretical support
5 Nonsmooth mechanics Stick-slip in friction
6 Non-smooth mechanics Unilateral contact Noise production (and reduction)
7 Articulated structures E.G. Floating platforms and bridges
8 Unilateral contact mechanism (α) without contact u >= 0 (β) with contact S >= 0 either (α) or (β), complementarity us=0
9 Complementarity Problems Linear Complementarity Nonlinear Complementarity
10 Unilateral Contact Interface Linear Complementarity Interface, Variational Inequalities, Nonsmooth Mechanics
11 Complementarity Problems (other applications) Flow in Networks (e.g. transportation) Equilibrium in the sense of Nash Walrasch Financial Mathematics Price analysis of derivatives and options Flow in porous media Multiphase flow, salt water front analysis etc
12 Monotone Laws and Convex Nonsmooth Superpotentials Unilateral contact
13 Monotone Laws Unilateral contact
14 Monotone Laws Unilateral contact
15 Monotone Laws Unilateral contact
16 Monotone Laws Unilateral contact
17 Variational Inequalities Semi-linearization, instead of linearization around a point of interest, which is based on Taylor expansion
18 Optimization with Constraints Normal and tangential cones, instead of linear subspaces in classical analysis
19 Nondifferentiable Optimization
20 Nonsmooth, convex optimization Characterization of minimum (instead of equalities, one solves differential inclusions or variational inequalities). Need for effective solution algorithms. Unilateral contact
21 Algorithms (convex) Unilateral contact
22 Smoothing methods Unilateral contact
23 Smoothing methods (formulation) Unilateral contact
24 Smoothing methods (results)
25 Optimization Convex Nonconvex
26 Nonmonotone laws from nonconvex superpotentials Unilateral contact Laws with falling branches
27 Non-monotone laws Delamination Nonclassical friction Shape Memory Materials Damage theories
28 One-dimensional example Unilateral contact
29 Two-dimensional Example Unilateral contact
30 Nonclassical Friction Law Stribeck Unilateral contact Important for the analysis of comlicated dynamic effects, selfexcited vibrations, chaotic effects, etc.
31 Nonsmooth Nonconvex Structural Analysis in Existing FEM
32 Unilateral Contact Interface (MARC) Equivalence: Mechanical Problem = Potential Energy Minimization Unilateral Contact: Minimization with Inequality Constraints Use of Augmented Lagrangian Method for Solution
33 Friction Modeling (general) Coulomb friction or Stribeck Model Using Slack Variables -> Resolution into LCP (cf. Previous Unilateral Contact Mechanism)
34 Friction Modeling (cont) Under assumptions: equivalence between mechanical problem and nonsmooth-nonconvex optimization problem
35 More general nonsmooth interfaces Example: adhesive crack element (available in some commercial software)
36 Friction modeling (MARC) Instead of non smooth Coulomb friction A smooth approximation is used
37 Elastoplastic Models (MARC) A first approximation for masonry material (without crushing) Better approach: no-tension (s-unilateral) material (nonstandard in FEM packages, must be introduced)
38 Other examples: Steel Structures (semirigid joints) Κ.Μ. Abdalla, G.E. Stavroulakis, C.C. Baniotopoulos
39 Semi-rigid Connections in Steel Structures Unilateral contact 2M5 or 3M5 or 4M5 UNSTIFFENED PLATE Load Case 3 2M5 or 3M5 or 4M mm MPa Design axis: X-X Mx=0.0 knm BS Bending Moment (knm/m) Bending Moments Moment max = 267 mm Distance (mm) K.M. Abdalla, G.E. Stavroulakis,
40 Subregions of the mesh which indicate the potential cracks with unilateral contact and friction Stavroulakis, Stavroulaki, Drosopoulos, Massalas
41 Static analysis due to weight and base settlement Load cases of base settlement: vertical movement 0.05m, vertical movement 0.10 m, vertical and horizontal movement 0.2m Stavroulakis, Stavroulaki, Drosopoulos, Massalas
42 Complementarity and variational inequalities Sources An introduction to complementarity Michael C. Ferris, Univ. of Wisconsin Optimization for modeling of nonlinear interactions, G.E. Stavroulakis Μη λεία και μη κυρτή ανάλυση, βελτιστοποίηση και μηχανική, Γ.Ε. Σταυρουλάκης
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