Iwona Skalna Department of Applied Informatics Cracow, Poland A method for outer interval solution of parametrized systems of linear equations Iwona Skalna Department of Applied Informatics Cracow, Poland NFS workshop on Reliable Engineering Computing 2004

NFS workshop on Reliable Engineering Computing 2004 Parametrized systems Consider a family of linear algeraic systems of the following type where and NFS workshop on Reliable Engineering Computing 2004

NFS workshop on Reliable Engineering Computing 2004 Solution set This family of linear algebraic systems is usually written in a compact form and is called parametrized system of linear interval equations. The solution set of such system is defined as NFS workshop on Reliable Engineering Computing 2004

NFS workshop on Reliable Engineering Computing 2004 New method Presented here method is based on the following inclusion where  denotes an interval hull, NFS workshop on Reliable Engineering Computing 2004

NFS workshop on Reliable Engineering Computing 2004 Algorithm NFS workshop on Reliable Engineering Computing 2004

NFS workshop on Reliable Engineering Computing 2004 Results In this presentation there is only one example shown, however in the paper some other examples may be found. The results produced by the described method are compared with the results of the method called Random Sampling of Parameter Intervals which gives an inner enclosure of the solution set. NFS workshop on Reliable Engineering Computing 2004

NFS workshop on Reliable Engineering Computing 2004 Example – plane truss y 30kN 30kN 30kN (0, 9) (4, 9) (8, 9) (12, 9) (16, 9) 1 4 2 3 5 8 6 7 9 (16, 6) (4, 6) (8, 6) (12, 6) 10 11 (16, 3) (12, 3) 12 (16, 0) x NFS workshop on Reliable Engineering Computing 2004

NFS workshop on Reliable Engineering Computing 2004 Example - description The exemplary plane truss structure is subjected to downward forces of 30 [kN] at nodes 2, 3, 4. All the bar elements have the same Young’s modulus E=7e10 [Pa], and the same cross-section area A=0.003 [m2]. The lengths (in meters) of the bars are shown in the figure. Now the stiffness of some of the bars (denoted in the figure with thick red lines) is assumed to be uncertain by 5%. NFS workshop on Reliable Engineering Computing 2004

Results – x coordinates RSPI rel. err.[%] Method dx2 -152.4 [-160.4, -145.2] 5 [-160.4, -144.4] 5.3 dx3 -228.6 [-236.5, -221.3] 3.3 [-236.6, -220.6] 3.5 dx4 [-163.7, -141.3] 7.4 [-165.3, -139.5] 8.5 dx5 -76.2 [-87.5, 65.2] 14.7 [-89.1, -63.3] 16.9 dx6 427.4 [419, 435.6] 1.9 [416.5, 438.3] 2.5 dx7 [417.5, 437.2] 2.3 dx8 351.2 [342.8, 359.4] 2.4 [341.3, 361.1] 2.8 dx9 dx10 267.9 [262.2, 273.6] 2.1 [261, 274.7] dx11 115.5 [109.8, 121.2] 4.9 [108.6, 122.3] 5.9 NFS workshop on Reliable Engineering Computing 2004

Results – y coordinates [x10-4] RSPI rel. err.[%] Method dy1 -308.2 [-312.5, -304.2] 1.3 [-315.1, -301.4] 2.2 dy2 -251.2 [-255.5, -247.2] 1.6 [-258.1, -244.5] 2.7 dy3 -149.8 [-152.8, -147.2] 1.9 [-153.4, -146.3] 2.4 dy4 -37.1 [-37.9, -36.4] [-38.2, -36.1] 2.8 dx5 4.3 dx6 1.7 dx7 -154.1 [-157.2, -151.4] [-158.3, -149.9] dx8 -32.9 [-33.5, -32.2] [-33.6, -32.1] 2.3 dx9 - dx10 -4.3 dx11 -24.3 [-25, -23.5] 3.1 [-25.2, -23.4] 3.8 NFS workshop on Reliable Engineering Computing 2004

NFS workshop on Reliable Engineering Computing 2004 Conclusions The problem of solving parametrized systems of linear interval equations is very important in practical applications. Well known classical methods (e.g. Gauss Elimination) fail since they compute enclosure for the solution set of nonparametrized systems which is generally much larger then solution set of the parametrized ones. A direct method for solving paraterized systems of linear interval equations was proposed and checked to be usefull in structure mechanics. The method produced tight enclosure for the solutions set of parametrized systems for exemplary truss structures. NFS workshop on Reliable Engineering Computing 2004