Mid-nineties can be considered the period when the main activities for uncertainty treatment in a form of intervals started in the area of Computational Mechanics and when the efforts for development of Interval Finite Element Methods has kicked off.

Koyluoglu, Cakmak and Nielsen (1995) have developed an interval approach utilizing finite element method to deal with pattern loading and structural uncertainties. The solutions for the system of linear interval equations obtained utilized the triangle inequalities and linear programming. The results were conservative bounds for the response quantities.

Köyluoglu, U., Cakmak, A.S., and Nielsen, S. R. K. (1995). "Interval Algebra to Deal with Pattern Loading and Structural Uncertainty, " Journal of Engineering Mechanics, November, 1149-1157.

Köyluoglu, U. and Elishakoff, I. (1998). "A Comparison of Stochastic and Interval Finite Elements Applied to Shear Frames with Uncertain Stiffness Properties, " Computers and Structures, Vol. 67, No. 1-3, pp.91-98.

Rao, S.S. and Sawyer, J.P. (1995), Rao, S.S. and Berke L (1997), and Rao, S.S. and Li Chen (1998) have developed different versions of interval based finite element methods to account for uncertainties in engineering problems, these works were mainly restricted to narrow intervals and approximate numerical results.

Rao, S. S., Sawyer, P. (1995). "Fuzzy Finite Element Approach for Analysis of Imprecisely Defined Systems, " AIAA Journal, Vol. 33, No. 12, pp 2364-2370.

Rao, S. S., Berke, L. (1997). "Analysis of Uncertain Structural Systems Using Interval Analysis, " AIAA Journal, Vol. 35, No. 4, pp 727-735.

A big effort has been devoted in the work of Rao, S.S. and Li Chen (1998) to develop a new algorithm for solution of linear interval equations, the algorithm used search-based operations with an accelerated step size with attempt to find an optimum setting of unknown vector components. In one of the examples they showed that 64 operations were needed to obtain the interval solution, where 128 operations are required by the combinatorial method (exact). That shows the inefficiency of the algorithm especially in large size problems, besides the sharpness of the results is limited to narrow intervals only.

Rao, S. S., LI Chen, (1998). "Numerical Solution of Fuzzy Linear Equations In Engineering Analysis, " Int. J. Numer. Meth. Engng. Vol. 43, pp 391-408.

Tonon and Bernardini (1996 &1999) have developed a mono and multi-objective optimization approach for structural uncertainty using fuzzy set and random set theories. In their work, they utilize a fuzzy set to model the engineer’s judgment about every objective function. Uncertainty was modeled by means of fuzzy numbers and the meaning was clarified by random set and possibility theories.

Bernardini, A. and Tonon, F. (1996). "A combined fuzzy and random-set approach to the multiobjective optimization of uncertain systems, " Proc, 7th ASCE EMD/STD Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, WPI, MA, August 7-9, 314-317.

Tonon, F. and Bernardini, A. (1999). "Multiobjective Optimization of Uncertain Structures Through Fuzzy Set and Random Set Theory, "Computer-Aided Civil and Infrastructure Engineering (previously Microcomputers in Civil Engineering), Vol.14.

Nakagiri and Yoshikawa (1996) have developed finite element interval estimation by convex model. The first-order approximation was employed to estimate the change of structural responses due to the change of structural parameters that are assumed to exist in a convex hull. The Lagrange multiplier method was employed to search for the bounds of the response.

Nakagiri, S. and Yoshikawa, N. (1996). "Finite Element Interval Estimation by Convex Model, " Proc., 7th ASCE EMD/STD Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, WPI, MA, August 7-9, 278-281.

Also, Nakagiri and Suzuki (1999) developed an interesting application of interval mathematics. In their work, they investigated the load identification as one of inverse problems of applied mechanics as well as the identification of material constants and boundary conditions.

Nakagiri, S. and Suzuki, K. (1999). "Finite Element Interval analysis of external loads identified by displacement input uncertainty, " Comput. Methods Appl. Mech. Engrg. 168, pp. 63-72.

Abdel-Tawab and Noor (1999), recently used parameters interval representation for analysis of dynamic thermo-elasto-viscoplastic damage response. The parameters number was limited; they used the combinatorial solution for the interval system to construct fuzzy membership of the response values.

Abdel-Tawab, K. Noor, A. K., (1999). "A Fuzzy-set Analysis for a Dynamic Thermo-elasto-viscoplastic Damage Response, ” Computers and Structures Vol. 70, pp 91-107.

Kulpa, Pownuk, and Skalna (1998), investigated possibilities to apply interval methods in analysis of linear mechanical systems with parameter uncertainties. This approach has been applied to trusses and frames.

Kulpa Z., Pownuk A., Skalna I., (1998) Analysis of linear mechanical structures with uncertainties by means of interval methods. Computer Assisted Mechanics and Engineering Sciences, vol. 5, pp.443-477.

Also Pownuk, A. (2001), has introduced an interval global optimization method. As stated in this work, the interval algorithm guarantees that all stationary global solutions have been found.

Pownuk A., (2001). New inclusion functions in interval global optimization of engineering structures. EUROPEAN CONFERENCE ON COMPUTATIONAL MECHANICS, Cracow, 26 - 29 June 2001, pp.460-461

Mullen and Muhanna (1995- 2001) have developed interval finite element analysis procedure utilizing the concept of fuzzy sets through interval calculations. They also have computed the response of different structural systems due to geometric and loading uncertainties. Results were exact in the case of load uncertainty where all types of dependency were eliminated in the new formulation. Exact bounds on possible nodal displacements and forces were calculated by combinatorial calculation of all loading patterns when computationally feasible.

Muhanna, R. L. and Mullen, R. L.(1995). "Development of Interval Based Methods for Fuzziness in Continuum Mechanics, " Proc., ISUMA-NAFIPS’95, September 17-20, 145-150.

Mullen, R. L. and Muhanna, R. L.(1996). "Structural Analysis with Fuzzy-Based Load Uncertainty, " Proc, 7th ASCE EMD/STD Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, WPI, MA, August 7-9, 310-313.

Muhanna, R. L. and Mullen, R. L. (1999). "Formulation of Fuzzy Finite Element Methods for Mechanics Problems, " Computer-Aided Civil and Infrastructure Engineering (previously Microcomputers in Civil Engineering), Vol.14, pp 107-117.

Mullen, R. L. and Muhanna, R. L (1999). "Bounds of Structural Response for All Possible Loadings, " Journal of Structural Engineering, ASCE, Vol. 125, No. 1, pp 98-106.

More recently, Muhanna and Mullen have developed an Interval Finite Element Method based on a new Element-By-Element (EBE) formulation for uncertainty in solid and structural mechanics. The method allows the treatment of uncertainty in a form of intervals (tolerances) for both load and stiffness terms. The present formulation avoids most sources of overestimation and computes a very sharp hull for the solution set of interval linear equations in the field of solid and structural mechanics, with wide interval quantities (exact in case of load uncertainty). The method allows calculating the very sharp bounds on the system response for all possible scenarios in a single analysis.

Muhanna, R. L. and Mullen, R. L., (2001). " Uncertainty in Mechanics Problems - Interval-Based Approach, "Journal of Engineering Mechanics, ASCE, Vol. 127, No. 6, pp 557-566.

Muhanna, R. L. and Mullen, R. L., (2001). " Treatment Of Geometric Tolerances In Finite Element Analysis," International Journal of Advanced Manufacturing Systems, Vol. 4, No. 1, pp 143-162.

Muhanna, R. L. and Mullen, R. L.(2000). "Sharp Enclosure for Material Uncertainty in Solid and Structural Mechanics-Interval Based Approach, Part I, " proceedings of the Eighth ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, University of Notre Dame, Notre Dame, Indiana July 24-26.

Muhanna, R. L. and Mullen, R. L.(2000). "Sharp Enclosure for Material Uncertainty in Solid and Structural Mechanics-Interval Based Approach, Part II, " proceedings of the Eighth ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, University of Notre Dame, Notre Dame, Indiana July 24-26.

Mullen, R. L. and Muhanna, R., L.(2000). "Sharp Interval Estimates for Finite Element Solutions with Fuzzy Material Properties, " proceedings of the 14th ASCE Engineering Mechanics Conference, The University of Texas at Austin, May 21-24.

Mullen, R. L. and Muhanna, R., L.(1999). "Interval-Based Geometric and Material Uncertainty for Mechanics Problems," proceedings of the 13th ASCE Engineering Mechanics Division Conference. The Johns Hopkins University, Baltimore, MD, June 13-16.

Muhanna, R., Ballarini, R. and R. Mullen (1998). "Calculation of Stress Intensity Factors with Fuzzy Loading, " proceedings of the 4-th International Conference on Stochastic Structural Dynamics, University of Notre Dame, Notre Dame, IN, USA, Aug. 6-8.

Mullen, R. L. and Muhanna, R. L.(1998). "Interval Based Finite Element Methods, " proceedings of the 4-th International Conference, NMA’98, Recent Advances in Numerical Methods and ApplicationsII, Aug. 19-23, 1998, editors, Iliev, O. P., Kaschiev, M. S., etc., World Scientific, 1999, pp 362-371.