A fuzzy finite element analysis technique for structural static analysis based on interval fields

Transcription

1 A fuzzy finite element analysis technique for structural static analysis based on interval fields Wim Verhaeghe, Maarten De Munck, Wim Desmet, Dirk Vandepitte and David Moens Katholieke Universiteit Leuven Department of Mechanical Engineering REC-2010 March 3rd, 2010

2 Outline 1 2 Interval Fields 3 4 : Vega Launcher 5

3 Outline non-determinism fields non-deterministic fields 1 Non-determinism Fields Non-deterministic fields 2 Interval Fields 3 4 : Vega Launcher 5

4 non-determinism fields non-deterministic fields Non-Determinism Causes of non-determinism: Variability Uncertainty (In)variable uncertainty and (un)certain variability Dealing with non-determinism: Probabilistic approaches: probability density function (PDF) Non-probabilistic approaches: Interval: x I = [x min x max] = [x x] Fuzzy: membership function µ x (x)

5 Fields non-determinism fields non-deterministic fields Field: Spatial coherence and distribution of a parameter. Input side of analysis: pressure distribution material properties Output side of analysis: deformation temperature In reality a strong dependency exists between the value of a parameter in one place and its value in a nearby place. How to represent this?

6 Non-deterministic fields non-determinism fields non-deterministic fields Dealing with non-deterministic fields at the output side: Probabilistic: Random fields, covariance function? Non-probabilistic: Interval vector {y I }, dependency? {y s } = {{y} ({x} {x I })({y} = f ({x}))} : interval vector representation of interval field 2D interval vector, introducing response surface

7 Outline 1 general concept static FE objectives 2 Interval Fields General concept Static Finite Elements Objectives 3 4 : Vega Launcher 5

8 Interval Fields General concept general concept static FE objectives Shapefunctions to describe a spatial pattern. [ψ] = [ 1 ψ 2 ψ...] Each pattern of [ψ] represents a possible fixed dependency between the components of the solution field. The non-determinism should influence the patterns and/or their contribution to the solution as a whole. It is clear that a realistic set of patterns is mandatory to describe the final results.

9 Interval Fields Static FE general concept static FE objectives Static finite element analysis: [K]{u} = {f} Deformation patterns for the output of a static finite element analysis: [ψ]{w} = {u} Projection of original set of equations on space of deformation patterns (classical static reduction) to find contribution of each pattern: [ψ] T [K][ψ]{w} = [ψ] T {f}

10 Interval Fields Static FE Influence of non-determinism: general concept static FE objectives Influence of variability/uncertainty {x} {x I } on: stiffness matrix: [K(x)] load vector: {f(x)} Results in: {u(x)} The deformation patterns provide an intermediate spatial dependency: [ψ(x)]{w} = {u(x)} Approximation of influence of non-determinism: Based on a few exact evaluations, the deformation patterns are approximated in the interval space: [ ψ(x)]{v} = {ũ(x)}

11 Interval Fields Static FE: objectives general concept static FE objectives Objectives: Determine optimal choice of spatial field basis at output of static structural IFE analysis: Based on static reduction Benefit: Smaller system of equations to be solved due to the projection of the original system on a few deformation patterns Using approximation for the basis in the interval space Benefit: Fast approximation of the dependencies using [ ψ(x)] Criterion: accuracy of static deformation results in complete interval space

12 Outline 1 deformation patterns approximation projection 2 Interval Fields 3 Deterministic Deformation Patterns Approximation of the patterns in the interval space Projection on the patterns 4 : Vega Launcher 5

13 Deterministic Deformation Patterns deformation patterns approximation projection The degrees of freedom are divided into subsets: c-set r-set l-set o-set Known, constrained dofs Dofs to be constrained to resist rigid body motion Dofs that are loaded Remaining dofs F c-set = t1 r-set = r1 l-set = t7 o-set = tr2, tr3, tr4, tr5, tr6, r7

14 Deterministic Deformation Patterns deformation patterns approximation projection Commonly used deformation patterns for static analysis: Rigid body deformation patterns: unit displacement for r-set Displacement deformation patterns: unit displacement for l-set Force deformation patterns: unit force/moment for l-set

15 Approximation The deformation patterns are approximated based on nominal and deviatoric parts. deformation patterns approximation projection { ψ(x)} = {ψ nom } + {ψ dev }f ( x) with x = x xnom x nom and f ( x) a function with a root for x = 0 Possible approximations: Based on three exact evaluations (nominal, maximal, minimal): quadratic: {ψ(x)} ={ ψ(x)} = {ψ nom} + {ψ linear } x + {ψ quadratic } x 2 exponential: {ψ(x)} ={ ψ(x)} = {ψ nom} + {ψ amplitude }(e x 1) + {ψ linear } x More advanced approximations can be constructed based on a higher number of exact evaluations.

16 Approximation example deformation patterns approximation projection dof 1 dof 2 dof 3 dof 4 dof 5 dof 6 X_min X_nom X_max X exact evaluation approximation

17 Projection dof 1 dof 2 dof 3 dof 4 dof 5 dof 6 [ ψ] T [K][ ψ]{w} = [ ψ] T {f} { ψ} = β{ψ nom } + γ{ψ dev }f ( x) deformation patterns approximation projection X_min X_nom exact evaluation approximation X_max X Classic projection: (β = γ) [ψ nom + ψ dev f ( x)] T [K(x)][ψ nom + ψ dev f ( x)]{v} = [ψ nom + ψ dev f ( x)] T {f(x)} Improved projection: (β γ) [ψ nom ψ dev f ( x)] T [K(x)][ψ nom ψ dev f ( x)]{v} = [ψ nom ψ dev f ( x)] T {f(x)}

18 Projection deformation patterns approximation projection Improved projection: Extensions Multiple uncertainties/variabilities: [ ψ(x)] = [ψ nom ψ dev1 f ( x 1 ) ψ dev2 f ( x 2 )...] Multiple dofs in l-set and multiple uncertainties/variabilities: [ ψ(x)] =[ 1 ψ nom 1 ψ dev1 f ( x 1 ) 1ψ dev2 f ( x 2 )... 2ψ nom 2 ψ dev1 f ( x 1 ) 2ψ dev2 f ( x 2 ) ]

19 Outline 1 input data results 2 Interval Fields 3 4 : Vega Launcher Input data Results 5

20 : Vega Launcher Description of part input data results copyright GW_Simulations, wikipedia courtesy of DutchSpace

21 input data results Subpart of the small launcher Vega: DOFS: 6726 Interval space is equidistantly sampled in 5 points, resulting in 5 5 = 3125 cases. Measure of error ndof n=1 error = u n ũ n n DOF u max Thickness uncertainties: Notation Min [mm] Nominal [mm] Max [mm] t t t t t

22 : Vega Launcher Results input data results x Projection with nominal and deviatoric parts kept seperated, mean error , standard deviation Projection with nominal and deviatoric parts added together, mean error , standard deviation

23 Outline 1 2 Interval Fields 3 4 : Vega Launcher 5

24 Using shape functions allows for the development of an interval field representation of uncertainties. Novel projection increases accuracy with little extra calculation costs. This representation provides a fast link between the input uncertainties and the output field for any possible interval technique. Extendable to any technique requiring multiple evaluations of a model influenced by non-determinism.

25 Questions Questions?