Interval Finite Element Research Articles

Interval identification of structural parameters using interval overlap ratio and Monte Carlo simulation

In this paper, a new interval finite element (FE) model updating strategy is proposed for interval identification of structural parameters in the aspect of uncertainty propagation and uncertainty quantification. The accurate interval estimation of system responses can be efficiently obtained by application of Monte Carlo (MC) simulation combined with surrogate models. By means of the concept of interval length, a novel quantitative index named as interval overlap ratio (IOR) is constructed to characterize the agreement of interval distributions between analytical data and measured data. Two optimization problems are constructed and solved for estimating the nominal values and interval radii of uncertain structural parameters. Finally, the numerical and experimental case studies are given to illustrate the feasibility of the proposed method in the interval identification of structural parameters.

An improved interval finite element method based on the element-by-element technique for large truss system and plane problems

The interval finite element method based on the element-by-element technique is proved to be a rigorous and efficient method for considering interval uncertainties. The structural internal force of the same accuracy as the interval response displacement can be obtained by combining the variational method and the Lagrange multiplier method. However, the increase of the Lagrange multiplier reduces the efficiency of solving the equation, especially for a large truss system. In addition, the method is mostly used to analyse truss system, and the applications in plane problems are still rare. In this article, an improved method involving modification of the Lagrange multiplier interval equations is proposed to calculate interval displacements and interval internal forces at the same time. In this proposed method, the increase of the unknown Lagrange interval does not affect the solution of the equations. For plane problems, a new method for solving the stress interval of the element is proposed. This proposed method can effectively reduce the expansion of the stress interval of the element. Finally, the correctness and rationality of the proposed method are verified by numerical examples.

Structural dynamic problems in time domain under uncertainty: an interval finite element approach

An analysis of the structural dynamic response under uncertainty is presented. Uncertainties in load and material are modelled as intervals exploiting the interval finite element method (IFEM). To reduce overestimation and increase the computational efficiency of the solution, we do not solve the dynamic problem by an explicit step-by-step time integration scheme. Instead, our approach solves for the structural variables in the whole time domain simultaneously by an implicit scheme using discrete Fourier transform and its inverse (DFT and IDFT). Non-trivial initial conditions are handled by modifying the right-hand side of the governing equation. To further reduce overestimation, a new decomposition strategy is applied to the IFEM matrices, and both primary and derived quantities are solved simultaneously. The final solution is obtained using an iterative enclosure method, and in our numerical examples the exact solution is enclosed at minimal computational cost.

Structural dynamic problems in time domain under uncertainty: an interval finite element approach

An analysis of the structural dynamic response under uncertainty is presented. Uncertainties in load and material are modelled as intervals exploiting the interval finite element method (IFEM). To reduce overestimation and increase the computational efficiency of the solution, we do not solve the dynamic problem by an explicit step-by-step time integration scheme. Instead, our approach solves for the structural variables in the whole time domain simultaneously by an implicit scheme using discrete Fourier transform and its inverse (DFT and IDFT). Non-trivial initial conditions are handled by modifying the right-hand side of the governing equation. To further reduce overestimation, a new decomposition strategy is applied to the IFEM matrices, and both primary and derived quantities are solved simultaneously. The final solution is obtained using an iterative enclosure method, and in our numerical examples the exact solution is enclosed at minimal computational cost.

Generalized reliability analysis of structues based on interval finite element

Relative to the special structure reliability theory only considering randomness, the advanced theory is called generalized reliability which considers randomness, fuzziness and unascertainty at the same time. This theory was first proposed by Wang GuangYuan academician in 1980s. In this thesis, some research results of the team of authors are introduced. Fuzziness is considered by the fuzzy mathematics and unascertainty is reckoned with the interval mathematics. First, some basic conceptions of interval math and interval finite element are introduced. Then generalized reliability analysis methods of three kinds of uncertainty conditions are discussed with the tool of interval finite element. In the first condition, both randomness and unascertainty are considered and the parameters of the random variables are interval variables. The second condition is that randomness and fuzziness are separated. The basic variables are random and the failure criterion is fuzzy. Randomness and fuzziness are coupling in the third condition. The parameters of the random variables are fuzzy numbers to reflect fuzziness. Finally, authors propose and develop the generalized reliability analysis methods of this three cases using the interval Monte Carlo Simulation (IntMCS) and the interval First Order Reliability Method (IntFORM). By means of a plane truss example, the applicability and effectiveness of these methods are shown. After a series of research, some conclusions can be summarized. The IntFORM proposed in this thesis has enough computational accuracy and more computational efficiency than IntMCS. Therefore, this method can be used in the reliability analysis of large-scale structures.

Euler–Bernoulli interval finite element with spatially varying uncertain properties

The formulation of an Euler–Bernoulli beam finite element with spatially varying uncertain properties is presented. Uncertainty is handled within a non-probabilistic framework resorting to a recently proposed interval field model able to quantify the dependency between adjacent values of an interval quantity that cannot differ as much as values that are further apart. Once the interval element stiffness matrix is defined, the set of linear interval equations governing the interval global displacements of the finite element model is derived by performing a standard assembly procedure. Then, the bounds of the interval displacements and bending moments are determined in approximate explicit form by applying a response surface approach in conjunction with the so-called improved interval analysis via extra unitary interval. For validation purposes, numerical results concerning both statically determinate and indeterminate beams with interval Young’s modulus are presented.

Analysis of wave motion in one-dimensional structures through fast-Fourier-transform-based wavelet finite element method

This paper presents a hybrid method that combines the B-spline wavelet on the interval (BSWI) finite element method and spectral analysis based on fast Fourier transform (FFT) to study wave propagation in One-Dimensional (1D) structures. BSWI scaling functions are utilized to approximate the theoretical wave solution in the spatial domain and construct a high-accuracy dynamic stiffness matrix. Dynamic reduction on element level is applied to eliminate the interior degrees of freedom of BSWI elements and substantially reduce the size of the system matrix. The dynamic equations of the system are then transformed and solved in the frequency domain through FFT-based spectral analysis which is especially suitable for parallel computation. A comparative analysis of four different finite element methods is conducted to demonstrate the validity and efficiency of the proposed method when utilized in high-frequency wave problems. Other numerical examples are utilized to simulate the influence of crack and delamination on wave propagation in 1D rods and beams. Finally, the errors caused by FFT and their corresponding solutions are presented.

Homogenization-based interval analysis for structural-acoustic problem involving periodical composites and multi-scale uncertain-but-bounded parameters.

This paper presents a homogenization-based interval analysis method for the prediction of coupled structural-acoustic systems involving periodical composites and multi-scale uncertain-but-bounded parameters. In the structural-acoustic system, the macro plate structure is assumed to be composed of a periodically uniform microstructure. The equivalent macro material properties of the microstructure are computed using the homogenization method. By integrating the first-order Taylor expansion interval analysis method with the homogenization-based finite element method, a homogenization-based interval finite element method (HIFEM) is developed to solve a periodical composite structural-acoustic system with multi-scale uncertain-but-bounded parameters. The corresponding formulations of the HIFEM are deduced. A subinterval technique is also introduced into the HIFEM for higher accuracy. Numerical examples of a hexahedral box and an automobile passenger compartment are given to demonstrate the efficiency of the presented method for a periodical composite structural-acoustic system with multi-scale uncertain-but-bounded parameters.

The vertex solution theorem and its coupled framework for static analysis of structures with interval parameters

SummaryThis work gives new statement of the vertex solution theorem for exact bounds of the solution to linear interval equations and its novel proof by virtue of the convex set theory. The core idea of the theorem is to transform linear interval equations into a series of equivalent deterministic linear equations. Then, the important theorem is extended to find the upper and lower bounds of static displacements of structures with interval parameters. Following discussions about the computational efforts, a coupled framework based on vertex method (VM) is established, which allows us to solve many large‐scale engineering problems with uncertainties using deterministic finite element software. Compared with the previous works, the contribution of this work is not only to obtain the exact bounds of static displacements but also lay the foundation for development of an easy‐to‐use interval finite element software. Numerical examples demonstrate the good accuracy of VM. Meanwhile, the implementation of VM and availability of the coupled framework are demonstrated by engineering example. Copyright © 2017 John Wiley & Sons, Ltd.

Non-stochastic interval factor method-based FEA for structural stress responses with uncertainty

The goal of this study is to evaluate behavior uncertainties of structures by using interval finite element analysis based on interval factor method as a specific non-stochastic tool. The interval finite element method, i.e., interval FEM, is a finite element method that uses interval parameters in situations where it is not possible to get reliable probabilistic characteristics of the structure. The present method solves the uncertainty problems of a 2D solid structure, in which structural characteristics are assumed to be represented as interval parameters. An interval analysis method using interval factors is applied to obtain the solution. Numerical applications verify the intuitive effectiveness of the present method to investigate structural uncertainties such as displacement and stress without the application of probability theory.

A substructure-based interval finite element method for problems with uncertain but bounded parameters

In this article, the dependency between different elements in solid structures is considered and a substructure-based interval finite element method is used to model the interval properties. The penalty method is applied to impose the necessary constraints for compatibility. In order to obtain the interval stresses, an approximation solution based on the Taylor expansion method is presented. Then, the proposed interval substructure model is expanded to nonlinear problems. In consideration of the nonlinear property of the elasticity modulus, an interval elastoplastic substructure analysis method using constant matrix based on the incremental theory is proposed and the interval expression of the interval stress updated formation is derived. Finally, numerical examples are carried out to demonstrate the reasonability and feasibility of the proposed method and evaluation system.

A novel Interval Finite Element Method based on the improved interval analysis

Static analysis of linear–elastic structures with uncertain parameters subjected to deterministic loads is addressed. The uncertain structural properties are modeled as interval variables with assigned lower bound and upper bound. A novel Interval Finite Element Method is formulated in the framework of the improved interval analysis via extra unitary interval, recently proposed to limit the conservatism affecting the classical interval analysis. The key idea of the novel method is to associate an extra unitary interval to each uncertain parameter in order to keep physical properties linked to the finite elements in both the assembly and solution phases. This allows one to reduce overestimation and perform standard assembly of the interval element matrices. The lower bound and upper bound of interval displacements and stresses are evaluated by applying two different strategies both based on the so-called Interval Rational Series Expansion for deriving the approximate explicit inverse of the interval global stiffness matrix. Numerical examples concerning 2D and 3D structures with uncertain Young’s modulus are presented to demonstrate the accuracy and efficiency of the proposed procedure.

Hybrid Chebyshev Interval Finite-Element and Statistical Energy Analysis Method for Midfrequency Analysis of Built-Up Systems with Interval Uncertainties

AbstractThe hybrid Chebyshev interval finite-element/statistical energy analysis (CI-FE/SEA) method is proposed as an improved method of mid-frequency response analysis for built-up systems with interval parameters, in which the Chebyshev series approximation theory is incorporated into the hybrid FE/SEA method. In the resultant hybrid CI-FE/SEA model, the uncertainty in the SEA subsystem is modeled by a nonparametric ensemble and dealt with hybrid FE/SEA equations analytically, whereas the uncertainty in the FE subsystem is modeled in terms of interval variables. Truncated Chebyshev series are used to approximate the average energy response and cross-spectrum response of the hybrid interval FE/SEA systems. The bounds of the truncated Chebyshev series with respect to the interval parameters are calculated through use of a Monte Carlo simulation. To compare the proposed hybrid CI-FE/SEA method with the conventional Taylor series based method, the hybrid second-order interval perturbation finite-element/sta...

Collocation interval analysis method for gust response

Latest aircraft design advances have started to recognize the important significance of defining multiple types of uncertainty. An important issue faced in the previous aeroelastic theory is how to deal with uncertain parameters in gust and structure. We describe the governing equation of the structural response of elastic wing in atmosphere due to gust excitation. The uncertain parameters describing the gust model and the wing structure are modeled as interval sets before the unified treatment of the uncertainties. A collocation interval analysis method( CIAM) for gust response based on the first Chebyshev orthogonal polynomials,interval collocation scheme and finite element method is proposed. The formula of CIAM is derived. The method does not require the sensitivities of the objective function with respect to uncertain variables and the assumption of narrow interval is also not needed. The proposed method can be used to solve gust response problems with uncertainties. Numerical example demonstrates that CIAM gives tighter gust response bound including exact response by comparing its results with Taylor interval analysis method( TIAM),which illustrates the efficiency and significant engineering value of the proposed method.

Hermitian Mindlin Plate Wavelet Finite Element Method for Load Identification

A new Hermitian Mindlin plate wavelet element is proposed. The two-dimensional Hermitian cubic spline interpolation wavelet is substituted into finite element functions to construct frequency response function (FRF). It uses a system’s FRF and response spectrums to calculate load spectrums and then derives loads in the time domain via the inverse fast Fourier transform. By simulating different excitation cases, Hermitian cubic spline wavelets on the interval (HCSWI) finite elements are used to reverse load identification in the Mindlin plate. The singular value decomposition (SVD) method is adopted to solve the ill-posed inverse problem. Compared with ANSYS results, HCSWI Mindlin plate element can accurately identify the applied load. Numerical results show that the algorithm of HCSWI Mindlin plate element is effective. The accuracy of HCSWI can be verified by comparing the FRF of HCSWI and ANSYS elements with the experiment data. The experiment proves that the load identification of HCSWI Mindlin plate is effective and precise by using the FRF and response spectrums to calculate the loads.

Nonlinear Finite Element Analysis of Frames Under Interval Material and Load Uncertainty

The present study focuses on the development of nonlinear interval finite elements (NIFEM) for beam and frame problems. Three constitutive models have been used in the present study, viz., bilinear, Ramberg–Osgood, and cubic models, to illustrate the development of NIFEM. An interval finite element method (IFEM) has been developed to handle load, material, and geometric uncertainties that are introduced in a form of interval numbers defined by their lower and upper bounds. However, the scope of the previous methods was limited to linear problems. The present work introduces an IFEM formulation for problems involving material nonlinearity under interval material parameters and loads. The algorithm is based on the previously developed high-accuracy interval solutions. An iterative method that generates successive approximations to the secant stiffness is introduced. Examples are presented to illustrate the behavior of the formulation. It is shown that bounding the response of nonlinear structures for a large number of load combinations under uncertain yield stress can be computed at a reasonable computational cost.

Interval Finite Element Analysis of Structural Dynamic Problems

We analyze the frequency response of structural dynamic systems with uncertainties in load and material properties. We introduce uncertainties in the system as interval numbers, and use Interval Finite Element Method (IFEM). Overestimation due to dependency is reduced using a new decomposition for the stiffness and mass matrices, as well as for the nodal equivalent load. In addition, primary and derived quantities are simultaneously obtained by means of Lagrangian multipliers that are introduced in the total energy of the system. The obtained interval equations are solved by means of a new variant of the iterative enclosure method resulting in guaranteed enclosures of relevant quantities. Several numerical examples show the accuracy and efficiency of the new formulation.

Uncertainty Analysis of Static Plane Problems by Intervals

We present a new interval-based formulation for the static analysis of plane stress/strain problems with uncertain parameters in load, material and geometry. We exploit the Interval Finite Element Method (IFEM) to model uncertainties in the system. Overestimation due to dependency among interval variables is reduced using a new decomposition strategy for the structural stiffness matrix and the nodal equivalent load vector. Primary and derived quantities follow from minimization of the total energy and they are solved simultaneously and with the same accuracy by means of Lagrangian multipliers. Two different element assembly strategies are introduced in the formulation: one is Element-by-Element, and the other resembles conventional assembly. In addition, we implement a new variant of the interval iterative enclosure method to obtain outer and inner solutions. Numerical examples show that the proposed interval approach guarantees to enclose the exact system response.

Response Bounds of Elastic Structures in the Presence of Interval Uncertainties

A mathematical-programming (MP)-based approach is proposed to directly determine the two extreme [(1) minimum, and (2) maximum] bounds of some static response quantity for elastic structures subject to the simultaneous application of interval applied forces and interval elastic material properties. Such a direct determination scheme reformulates the interval finite-element (FE) analysis into a pair of standard nonlinear programming (NLP) problems that can be solved by any available NLP code. Not only does the proposed method advantageously bypass any computationally expensive combinatorial searches to capture the response limits, but it can also directly accommodate any dependency in the interval data. Some examples are presented to illustrate the robustness and efficiency of the method.

Geometric misfitting in trusses and frames: an interval-based approach

In this work, the issue of predicting structural behaviour in the presence of geometric uncertainty arising out of fabrication errors and/or thermal changes in trusses and frames is addressed. Geometric uncertainty is expressed as deviations of actual dimensions of system components from their corresponding nominal dimensions (misfitting) and is expressed in interval form. Such geometric uncertainty is converted into an equivalent nodal load uncertainty. The present work eliminates the element force overestimation resulting in the previous formulation of Muhanna et al. The mixed interval finite element formulation developed earlier by Rama Rao et al. is utilised to obtain sharp bounds to displacements and element forces in the presence of geometric, material and load uncertainties. The present work also computes an exact enclosure to the final system geometry. Results are illustrated in example problems. Dependency due to the presence of geometric uncertainty is eliminated using the M matrix approach developed earlier by Mullen and Muhanna.