FSD and its Modifications for Shell Finite Elements

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From this equation we obtain well-known iterative relation (9). Mentioned geometric interpretation shows, that new estimation is approximated from points [0, 0] and []. From the numerical mathematics point of view we are speaking about Regula-Falsi method (method of chords-secants) ^{[8, 11]}. FSD iteration relation can be derivedfrom these equations as follows:

(9)

Where is predicted stress for thickness in (k+1)^th step, is stress in k^th step tor thickness , is thickness in k^th step, isn^th thickness from vector of allowed thicknesses.

It means if then t_k+1 is “the best value from t_n values” for condition

3.2. 2^nd Algorithm - "Squared" FSD Method

This method originated from the "classical" approach (Alg. No. 1). Previous numerical tests proved instability of algorithm No. 1 and an interesting behavior was observed. The "classical" approach seemed to oscillate and slowly approach the optimum, the goal of this improved so called "squared" FSD method is to smooth the progress of changes of new variable values, thus minimize oscillation and accelerate convergence to the optimum ^[9]. The predicted equivalent stress is obtained by formula

(10)

Figure 2. Geometrical representation of algorithm No. 2

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Despite similarity between formula used in this and the one used in the previous algorithm, the addition of the second exponent proved to make significant influence on the results. Numerical tests showed high stability when optimizing models with higher number of optimization groups while preserving reasonable speed.

3.3. 3^rd Algorithm - Newton’s Methodforpredictionof Equivalent Stress

This method is very similar to Newton's tangent method ^{[6, 8, 9]}. The problem of acquiring derivation of an unknown relation between thickness and equivalent stress of particular optimization group was solved by Newton numerical algorithm. As we lack preceding results (from (k-1)^th step) in the first iteration, we have to use another method to acquire the first set of results and the vector of thicknesses used in the second analysis.

Geometric interpretation of the method is shown in Figure 3. Starting from similarity of triangles ABC and BDE we will get

(11)

and after modification we obtain following iterative relation

(12)

However, as stated before, we based our methods on stress prediction, so we used following iterative relation for prediction of von Mises equivalent stress. Predicted stress in (k+1)^th iteration for thickness t_n is calculated as follows:

(13)

Where is predicted stress for thickness t_n in (k+1)^th step is stress in k^th step for thickness , is thickness in k^th step, = n^th thickness from vector of allowed thicknesses, isthickness in (k-1)^th step, is stress in (k-1)^th step.

Figure 3. Geometrical representation of algorithm No. 3

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This method proved to be very fast while dealing with relatively small number of variables, although with larger number of variables stability problems can arise preventing successful finding of solution.

3.4. 4^th Algorithm - Linear Approximation of Internal Forces and Moments

The fourth proposed algorithm is using linear approximation to predict membrane forces and bending moments for thickness t_nassuming linear relation between force quantity and inverse of thickness. Numerical tests proved this method to be reliable, without stability problems and preserving speed while optimizing both small and large number of optimization groups. Predicted membrane forces and bending moments are

(14)

and

(15)

Predicted stresses on the top and bottomcan be expressedas follows

(16)

(17)

Figure 4. Geometrical representation of algorithm No. 4

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Predicted equivalent stress is calculated using von Mises hypothesis (Eq. 4).

3.5. 5^th Algorithm-Newton’smethod for Prediction of Internal Forces and Moments

An approach similar to the third algorithm was implemented for prediction of internal forces and moments.Geometric interpretation of the method is shown in Figure 5. Following relations are based on similarity of triangles ABC and BDE.Predicted membrane forces and bending moments will be

(18)

(19)

Figure 5. Geometrical representation of algorithm No. 5

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Predicted stresses are calculated using (9), (10) and (4).

3.6. 6^th Algorithm-“Squared" Approximation of Internal Forces and Moments

As addition of second exponent in the algorithm No. 2 proved to be significant improvement to "classical" approach of the FSD method, authors also tested similar change to algorithm No. 4 (linear approximation of internal forces) and added square root to relationship

Figure 6. Geometrical representation of algorithm No. 6

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Predicted membrane forces and bending moments:

(20)

(21)

Predicted stresses are calculated using equations (9), (10) and (4).

3.7. 7^th Algorithm-"Mixed" Approximation of Internal Forces and Moments

Algorithm No. 7 is a combination of linear approximation of membrane forces from algorithm No. 4 and nonlinear approximation of bending moments from algorithm No. 6.

Predicted membrane forces and bending moments:

(22)

(23)

Predicted stresses are calculated using (9), (10) and (4).

4. Testing Example

Presented seven computational algorithms were tested and compared on cranked beam Figure 1. The 3-node triangular shell elements were used. Number of finite elements was 5462 and number of nodes was 5739.The material model was linear elastic isotropic steel withYoung's modulus of elasticity E=2,1.10⁵MPa, Poisson's ratioν=0,3, stress limit Following boundary conditions were used:

- load conditions: pressure p = 1.0 MPa on three quarters of upper flange,

- deformation conditions: fixed left side.

Figure 7. Model geometry and boundary conditions

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Figure 8. Optimizationgroups: No. 1 - red, No. 2 - green, No. 3 - blue, No. 4 - cyan, No. 5 - magenta

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5. Results

The results of analyses present means of improving the classical formulation of the FSD method. It has to be noted that the increase of effectiveness is not as significant as authors expected. Summary of iteration steps needed is presented in Table 1 and Figure 9. History of mass convergence of individual algorithms is in Figure 10. The Figures. 11 until 17 present the history of convergence of the equivalent stresses represented each optimizing group.

Table 1. Summary of iteration steps number

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Tables index

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Figure 9. Number of algorithm used vs. number of iteration steps needed for solution

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Figure 10. History of mass convergence for algorithm 1 to 7

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Figure 11. History of stress convergence for 1^st algorithm

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Figure 12. History of stress convergence for 2^nd algorithm

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Figure 13. History of stress convergence for 3^rd algorithm

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Figure 14. History of stress convergence for 4^th algorithm

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Figure 15. History of stress convergence for 5^th algorithm

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Figure 16. History of stress convergence for 6^th algorithm

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Figure 17. History of stress convergence for 7^th algorithm

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