Neuro -Finite Element Static Analysis of Structures by Assembling Elemental Neuro -Modelers

Neuro -Finite Element Static Analysis of Structures by Assembling Elemental Neuro -Modelers Abdolreza Joghataie Associate Prof., Civil Engineering Department, Sharif University of Technology, Tehran, Iran. Joghatae@sharif.edu 1 Introduction Abstract Recently, several algorithms have been proposed for using neural networks in dynamic analysis of small structural systems, and also constructing adaptive material modeling subroutines with the aim of their implementation in finite element computer programs. In these algorithms, the neural networks are trained based on the data obtained from tests at structural or material levels. In this paper, a new method of using neural networks for static analysis of non-linear structures is presented which works with test results at elemental level. The method is based on the assemblage of elemental information to form the global information about the structure, and hence it is called the Neuro-Finite Element (NFE) method. Multi-layer feed-forward neural networks are used. Two small sample problems, a plane truss and a plane frame, and also a larger space truss are included for illustration. The results have been promising. Keywords: Neuro-finite element, Static, Analysis, Structures, Neuro-modelers. Two main characteristics of the available algorithms for the nonlinear finite element analysis of structures are that: (1) they work based on mathematical models which explain the material behavior or the stiffness matrices of the elements of a structure, where such mathematical models are based on test results at material or elemental levels. Tests at material level and finding material models is generally easier and less expensive than tests at elemental level; however using material models, stiffness matrices of the elements of a structure are obtained through integration which provides less precise results than finding the stiffness of the elements directly from tests at elemental level. (2) they are incremental, and hence their convergence to the exact solution of a problem requires a large number of computational steps. It is not easy to find mathematical models for many structural materials or elements such as composites including concrete, fiber reinforced concrete and steel-concrete beams and columns. Also t he required large number of computational time steps of analysis is not desirable. Some attempts have been made recently to overcome these two difficulties. For example some researchers have used neural networks to extract the rules governing the mechanical behavior of some structural materials. To this end Ghaboussi et al. [1] and Wu [2] have used neural networks for modeling the stress-strain relationship of plain concrete under cyclic loading, Ellis et al. [3] have used them to model stress-strain behavior of sands, Pidaparty and Palakal [4] have modeled the stress-strain relationship for graphiteepoxy laminates under monotonic and cyclic loading, Ghaboussi et al. [5], [6] have proposed a method of designing more sophisticated neural networks for modeling composite materials, Kaklauskas and Ghaboussi [7] have designed neural network models for concrete based on flexural test results of reinforced concrete members with various depth, reinforcement ratio and rebar size, and finally Haj-Ali et al. [8] have used them for preparing micro-mechanical models for heterogeneous

materials. Also to reduce the computation cost, a new flexibility based FE method called the Large Increment Method (LIM), which does not use neural networks, has been proposed by Aref and Guo recently [9]. Neural networks have been used in modeling the dynamic response of actively controlled structures too. Ghaboussi and Joghataie [10] and Bani-Hani and Ghaboussi [11] have proposed neural network based predictive control algorithms for the control of linear and non-linear frames respectively. Such modeling requires dynamic tests at structural level which are technically much harder than tests at material and element levels. In fact, the corresponding neural networks, which are called neuro-emulators, can not be considered as analysis tools in that they can only predict a limited number of the future time steps of the response and also they require to use, in their input layer, the measured response by the sensors. In this paper, a new method of using neural networks for static analysis of nonlinear structures, called the Neuro-Finite Element (NFE) method is presented which, instead of using the results of loading a structure or preparing material neuromodels for finite element analysis, is bas ed on the use of the results of experiments on the elements of the structure. The method is called neuro-finite element in that, like the conventional finite element method, it assembles the information at the elemental level in order to construct another neural network which can represent the stiffness relationship between the vectors of nodal forces and nodal displacements of the discretized structure. The advantage of this method over the other neural network based analysis methods is in that it works based on testing at elemental level which is much more easier than testing a structure and more precise than using material models. In this paper, first the Neuro-Finite Element (NFE) method and computational algorithm will be presented. Then for illustration, a 2-D truss, a 2-D frame and a 3-D space truss will be analyzed by the NFE method and the results are compared with the exact results. Multi-Layer Feed-Forward s (MLFFNN) are used in this study. Because of space limitation, MLFFNNs and their training procedure are not explained here; and it is assumed that the interested reader is either familiar with MLFFNNs or will refer to appropriate references if necessary. Some of the relevant references are [12] and [13]. 2 Neuro-Finite Element (NFE) Algorithm The algorithm is developed for time-independent static structures with one-dimensional elements such as trusses and frames. Only material non-linearity is considered here. The steps are: 1) Discretize the structure and define: The force and displacement vectors, R=[R1, R2,,R n] T and u=[u1, u2,,un] T, where n= number of degrees of freedom of the structure; The displacement vector of element i, q i,,where i=1,2,,m, and m= number of elements of the structure; The internal force vector of element i, Q i,, where i=1,2,,m; The global vectors q*= [q 1 T, q 2 T,, q m T ] T and Q*= [Q 1 T, Q 2 T,, Q m T ] T ; 2) Find the displacement transformation matrix a, where q* = a u (1) noticing that from the principle of virtual work R = a T Q* (2) 3) Train a MLFFNN for each element i=1, 2,, m, to learn the q i Q i mapping. Call it the elemental displacement-force neural network of element i, denoted by (EDFNN) i. The training data can be obtained from experiments. 4) Find the relationship between u and R using EDFNNs, according to the following algorithm: - Generate a number of vectors u j A n, j=1, 2,, nu, where A= the set of admissible values of u which satisfy the assumption of small displacements so that the linear Equations (1) and (2) are valid. Find from Equation (1), q* j = a u j j=1, 2,, nu (3) - Extract q i,j, i=1,2,, m, and j=1, 2,, nu, from q* j. - Find Q i,j, corresponding to q i,j, by using the (EDFNN) i, for i=1, 2,, m, and j=1, 2,, nu.

- Construct according to step (1), Q* j = [Q 1,j T, Q 2,j T,, Q m,j T ] T, j=1, 2,, nu (4) - Find R j from Equations (2) and (4) as: R j = a T Q* j, j=1, 2,, nu (5) - Train a MLFFNN, called the neuro-analyzer or more specifically the force-displacement neural network denoted by FDNN, to learn the R j u j mapping, for j=1,2,,nu. The FDNN can now be used for the analysis of the structure. Given any force vector R, it estimates the corresponding displacement vector u. 3 Illustrative Examples Three sample problems are solved by the NFE method. A small plane truss, Fig. 1a, a small plane frame, Fig. 1b, and a larger space truss, Fig. 1c. The first example is explained in more details but because of page limitations, only some of the results for the plane frame and the space truss problems are included. 3.1 Plane Truss From Fig. 1a, ndofs=4. The elements are assumed to be made up of non-linear materials of different properties. Following NFE algorithm, 1) u=[u 1, u 2,,u 4 ] T, m=5, R= [R 1, R 2, R 3, R 4 ] T, q* = [q 1, q 2,, q 5 ] T, Q*= [Q 1, Q 2,, Q 5 ] T tensile (+) compressive (-), 2) a is: 0 1 0 0-1 0 1 0 a= 0 0 0 1, (6) 0 0 0.8 0.6-0.8 0.6 0 0 3) An EFDNN is trained for each of the elements. For this numerical study, it is assumed that the real behavior of each element, can be described by: dq i / dq i = [ (K 1,i - K 2,i ) exp( α i q i 2 ) + K 2,i ] (7) for i=1,2,,5,where dq i / dq i = first derivative of Q i with respect to q i,k 1,i and K 2,i = the initial and final tangent stiffness of element i respectively, and α i = a material constant corresponding to the material of element i. In the above equation, qi, Qi, K1,i, K2,i and α i are in cm, kn, kn/cm, kn/cm and 1/cm 2 respectively. This form of material nonlinearity was selected because it is similar to the behavior of many structural materials. r 2, R 2 r 4, R 4 B b C r 1, R 1 r 3, R 3 a c d A D 4 m e 3m r2,r2 r3,r3 r 1,R 1 B b C a c 3m A 1 1 4 x 5m Node Position (m) z 1 ( 0,-2, 0) 2 ( 2, 2, 0) 7 3 (-3, 3, 0) 6 4 ( 0,-1, 3) 5 ( 1, 1, 3) 6 (-1.5,1.5,3) 4 5 6 9 7 ( 0, 0, 6) 3 8 12 7 3 5 11 10 2 (c) 2 Figure 1: Illustrative examples In this example, the initial stiffness of the members is calculated based on the modulus of elasticity of steel. Table 1 contains the values chosen for this example. According to Table 1 and Equation (7) the behavior of each element is different from the other elements. D y

Table 1: Structural Properties of Truss Elements Properties Element K 1,i K 2,I α i Number, i kn/cm kn/cm 1/cm 2 1 4000 2100 0.35 2 3500 2000 0.7 3 3800 1800 1.2 4 4500 2500 1.4 5 4200 2000 1.5 Training of the EFDNNs: Each q i, i=1,2,..,5, was changed from 2 cm to +2 cm with increments of 1 cm. Assuming Q i =0 at q i =0, the (Q i, q i ) points were obtained by summing the increments of?qi calculated from Equation (7). So the number of q-q training pairs =nu=401. The EFDNNs were trained incorporating both linear and non-linear neurons. Fig. 2a shows the architecture of the EDFNNs. All of them have the same architecture but with different connection weights. Sigmoidal Nodes: f(x)=-1+2/(1+exp(-1.2 x)) Linear Nodes: f(x)=x q i Q i R u Figure 2: Architecture of EFDNNs and FDNN for the plane truss EFDNNs, FDNN Testing the EFDNNs: To test the capability of the EFDNNs both in interpolation and extrapolation, 801 values of q i, i=1,2,..,5, with increments of 1 cm, in the range of [-4,4] cm, were generated and fed to the corresponding EFDNNs. Fig. 3 shows the estimated and exact Q 3 as a function of q 3. The part of the curve where q 3 =[-2, +2] cm, is related to the interpolation while the rest of the curve shows the extrapolation the capability of (EDFNN) 3. It is seen that extrapolation degrades very slightly as q i gets farther from the training range. Similar results were obtained for the other members too. Training the FDNN: Following step 4 of the NFE method, 10000 displacement vectors were generated randomly by assigning random numbers in the range of [-2,+2] cm to each of the 4 components of the u vector, hence nu=10000. Using the EDFNNs, the corresponding R vectors were calculated. The obtained R-u pairs were then used in the training of the FDNN for the truss. Fig. 2b shows the final architecture of the FDNN. 10000000 8000000 6000000 Exact answers 4000000 2000000 0 (0,0) -2000000-4000000 -6000000-8000000 -10000-10000000 q (cm) -4.00 0 0 0 0 0 0 0 4.00 Q(kN) Figure 3: EFDNN 3 estimations of Q 3 - q 3 Testing the analysis capability of the FDNN: 100 force vectors, R were generated randomly and fed to the FDNN, and the corresponding output vectors, u, were recorded. The accuracy of results were checked for each loading case, following Equations (1) and (2). Fig. 4 compares the analysis results by the u1 (cm) u2 (cm) u3 (cm) u4 (cm) (c) 1 11 21 31 41 51 61 71 81 91 101 (d) Test Case Number Figure 4: Comparison of analysis results by FDNN with exact results, for plane truss problem FDNN with the exact results, where the displacements of each of the 4 degrees of freedom are plotted for all of the 100 test cases. The estimation error is negligible in structural engineering applications.

3.2 Plane Frame As shown in Fig. 1b, the frame has 3 members and 3 degrees of freedom, 1 side sway and 2 nodal rotations. The members of the frame were considered to work in flexure only, assuming their axial deformation is negligibly small. Each element of the frame has a displacement vector corresponding to two end rotations as its degrees of freedom, q i =[q i,1, q i,2 ] T, i=1,2,3. The stiffness of the elements were assumed to follow: dq i,1 /de i,1 =[(K 1,i,1 -K 2,i,1 )exp( α i,1 ( e i,1 ) 2 )+K 2,i,1 ] (8) dq i,2 /de i,2 =[(K 1,i,2 -K 2,i,2 )exp( α i,2 ( e i,2 ) 2 )+K 2,i,2 ] (9) where e i,1 =(2q i,1+ q i,2 ) and e i,2 = (q i,1+ 2q i,2 ) (10) and i= element number= i=1,2,3. Q i,1 and Q i,2 = end moments of the element, K 1,i,1 and K 2,i,1 = initial and final stiffness corresponding to end number 1 of element i, and K 1,i,2 and K 2,i,2 = initial and final stiffness corresponding to end number 2 of element i; and α i,1 and α i,2 = material constants corresponding to the two ends of element i. In Equations (8) and (9), q, Q, K, and α are in radians, MN-cm, MN-cm and 1/(radians 2 ). Table 2 contains the values chosen for this theoretical example. Table 2: Structural Properties of Frame Elements Properties Element K 1,i,1 K 2,i, 1 α i,1 K 1,i, 2 K 2,i, 2 α i,2 Number, i MN-cm MN-cm 1/(rad) 2 MN-cm MN-cm 1/(rad) 2 1 1500 800 2000 1300 700 1500 2 1750 1000 3000 1450 900 2500 3 1200 900 6000 1650 1000 4500 Following the NFE steps, 3 EDFNNs were trained for the elements of the frame and tested. All of the EFDNNs posessed linear input and output nodes, and one hidden layer with 3 linear and 5 sigmoidal nodes; [2,8,2]. 10000 u vectors were generated randomly and used in the training of the FDNN. The FDNN with linear input and output nodes and one hidden layer with 3 linear and 5 sigmoidal nodes, [3,8,3], was then tested on 100 randomly generated load vectors R as was done for the truss example. Comparison of the predicted and exact results are shown in Figs. 5. As can be seen, the NFE method has been successful in providing appropriate results. u2 (radians) u3 (radians) u1 (cm) 04 02 00-02 -04 04 03 02 01 00-01 -02-03 -04 1 11 21 31 41 51 61 71 81 91 101 (c) Test Case Number Figure 5: Comparison of analysis results by FDNN with exact results, for plane frame problem 3.3 Space Truss As shown in Fig. 1(c), the space truss has 7 nodes and m=12 elements. To each of the free nodes, 3 degrees of freedom are associated, hence n= 12. It is desired to find the vertical displacement of the top node, u 12, for any arbitrary loading on the truss. For this numerical study, it is assumed that the cross sectional areas of the elements are selected so that 4 different elemental force-displacement relationships could be considered for the elements, as: Q=θ (e αq -β e -αq )/(e αq + e -αq ) +θ (β-1)/2 +ηq (14) where θ, α,β and η represent the material constants. The constants for each of the elements are reported in Table 3. R and u are 12 dimensional. Also q* and Q* are 12 dimensional. Hence a is a 12 12 matrix. In this research, is was also desired to study the effect of the size of the training pool on the accuracy of the FDNN estimations. Hence two neural networks were trained based on 1000 and 10000 randomly generated training cases. To this end 10100 u= [u 1, u 2,,u 12 ] vectors, with u i [-2,+2] were generated randomly, and their corresponding q*, Q*, and R vectors were calculated. The first 10000 u- R pairs were kept for the training of the two FDNNs, while the remaining 100 were used in their testing.

Table 3: Properties of elements of space truss Material Constants Material θ α β γ Elements Number kgf=10n 1/cm (kgf=10n)/cm Number 1 20000 5 0.6 300 1,2,3 2 30000 4.9 0.55 450 4,5,6 3 35000 5.1 0.65 150 7,8,9 4 28000 3.8 0.6 390 10,11,12 The dotted and solid curves in Fig. 6 correspond to the error of testing of the two neural networks respectively. As can be seen the results are close though using 10000 training cases has provided better results. As can be seen, the errors in both cases are, from the viewpoint of practical structural engineering, so small that the estimated and true displacements could not be distinguished when they were drawn on the same figure. For space saving, this figure is not included here. ERROR (cm) 07 06 05 04 03 02 01 - - - - 1000 Training Cases 10000 Training Cases 0 0 20 40 60 80 100 TEST CASE NUMBER Figure 6: Effect of size of training pool on precision of FDNN, for space truss example 4 Conclusions It was shown in this paper that the proposed NFE method can be successful in analysing nonlinear truss and frames of small and moderate size. However its application to larger structures requires more work on the subject. Acknowledgement This research was partially supported by the Research Deputy of Sharif university of Technology, Tehran, Iran. This support is greatly acknowledged. References [1] J. Ghaboussi, J. H. Jr. Garrettr, X. Wu. (1991). Knowledge-based material modeling with neural networks. ASCE, J. Engrg. Mech., 117(1), pages 132-153. [2] X. Wu (1991). Neural network based material modeling. PhD. Thesis, University of Illinois at Urbana-Champaign, Urbana, IL. [3] G. Ellis, C. Yao, R. Zhao, D. Penumadu (1995). Stress-strain modeling of sands using artificial neural networks. ASCE, J. Geotechnical Engrg., 102, pages 429-435. [4] R. M. V. Pidaparty, M. J. Palakal (1993). Material models for composites using neural networks. AIAA J., 31 (8), pages 1533-35. [5] J. Ghaboussi, D. A. Pecknold, M. Zhang, R. Haj-Ali (1996). Neural network constitutive models determined from structural tests. Proc. 11 th ASCE Engrg. Mech. Conf., May 1996. [6] J. Ghaboussi, D. A. Pecknold, M. Zhang, R. HajAli (1998). Autoprogressive training of neural network constitutive models. Int. J. Num. Meth. in Engrg., 42 pages 105-126. [7] G. Kaklauskas, J. Ghaboussi (2001). Stress-strain relations for cracked tensile concrete from RC beam tests, ASCE J. Struct. Engrg., 127(1), pages 64-73. [8] R. Haj-Ali, D. A. Pecknold, J. Ghaboussi, Z. Voyiadjis (2001). Simulated micromechanical models using artificial neural networks. ASCE J. Engrg. Mech., 127(7), pages 730-38. [9] A. J. Aref, Z. Guo (1998). Framework for finiteelement-based large increment method for nonlinear structural problems. ASCE J. Engrg. Mech., 127(7), pages 739-746. [10] J. Ghaboussi, A. Joghataie (1995). Active control of structures using neural networks. ASCE J. Engrg. Mech., 121(4), pages 555-67. [11] K. Bani-Hani, J. Ghaboussi (1998). Nonlinear structural control using neural networks. ASCE J. Engrg. Mech.,124 (3), pages 319-327. [12] S. Haykin (1999). s: a Comprehensive Foundation. 2 nd edition, Prentice-Hall international. [13] A. Joghataie, J. Ghaboussi, X. Wu (1995). Learning and Architecture determination through automatic node generation. Proc. intelligent engrg. systems through artificial neural networks, ASME 1995, pages 45-50.