Related Information. Close Figure Viewer. Browse All Figures Return to Figure. Previous Figure Next Figure. Email or Customer ID. Forgot password? Old Password. New Password. Password Changed Successfully Your password has been changed. Returning user. The simulation was performed for every frequency level in the input domain, based on 10, samples. Only samples were necessary to emulate the statistics of the frequency response at a comparable accuracy.

Conclusions The stochastic vibration response of a randomly parametrized structural dynamic system has been considered in this pa- per. Then the evaluated response at these random samples provide training runs which are used by a Bayesian metamodel, which emulates the system response in order to estimate the uncer- tainty distribution and derive response statistics across the frequency range.

Higher order spectral functions produce a more accurate approximation of the stochastic system response but are com- putationally more expensive. A Bayesian emulator can be used to reduce the cost associated with an increasing order of spectral functions. Samples drawn from the posterior distribution can be used to perform uncertainty analysis of the response. A corrugated panel with random elastic parameters has been analyzed with the proposed approach.

This might contribute to the understanding of the dynamic behavior of this type of structure and their use in future applications of morphing aircraft. The response curves obtained with the spectral function and the direct MCS method are in good agreement even near the resonance frequencies. The relative error plots show an increase in the accuracy of the approximated solution when using higher order spectral functions.

A statistical summary of the system response, such as the mean or the standard deviation, can be emulated, whenever a Monte Carlo estimate is not feasible. The applicability of the method stems from the enhanced accuracy of the approximate solution obtained due to the use of carefully chosen stochastic preconditioners. Future work along this direction may extend the underlying idea of the methodology presented here to the case of tran- sient time domain response of stochastic dynamical systems. For example, the proposed spectral approach can be used at every linearization step of the iterative technique required in the response evaluation.

While the increase in spectral order increases the accuracy of the approximated solution, the choice of the optimum order of spectral functions is not obvious from the relative error analysis results presented here and there is scope to introduce adaptivity, based on some a posteriori error estimators. References [1] H. Papadrakakis, V. Pradlwarter, G. Non-linear Mech. McKay, W. Conover, R. Beckman, A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics 21 2 — Kleiber, T.

Yamazaki, M. Shinozuka, G. Matthies, A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Comput. Pettit, P. Beran, Spectral and multiresolution wiener expansions of oscillatory stochastic processes, J. Sound Vibr. Ghanem, P. Xiu, G. Safety 91 10—11 — Sacks, W. Welch, T. Mitchell, H. Wynn, Design and analysis of computer experiments, Stat.

Santner, B. Williams, W. DiazDelaO, S. Adhikari, Structural dynamic analysis using Gaussian process emulators, Eng. Saavedra Flores, F. DiazDelaO, M. Friswell, J. Methods Eng. Press, Tempone, G. Ghanem, D. Adhikari, Joint statistics of natural frequencies of stochastic dynamic systems, Comput. Adhikari, Calculation of derivative of complex modes using classical normal modes, Comput.

Nair, A. Pichler, H. Schueller, A mode-based meta-model for the frequency response functions of uncertain structural systems, Comput. Goller, H. Schuller, An interpolation scheme for the approximation of dynamical systems, Comput.

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Falsone, G. Ferro, An exact solution for the static and dynamic analysis of FE discretized uncertain structures, Comput. Sarkar, R. Ghanem, Mid-frequency structural dynamics with parameter uncertainty, Comput. Ghanem, R. Keese, H. Blatman, B. Wan, G. Karniadakis, An adaptive multi-element generalized polynomial chaos method for stochastic differential equations, J.

Grimme, Krylov projection methods for model reduction Ph. Kozyakin, On accuracy of approximation of the spectral radius by the Gelfand formula, Linear Algebra Appl. Oakley, A. B 66 3 — Haylock, A. Oakley, Eliciting Gaussian process priors for complex computer codes, The Statistician 51 1 81— Rougier, Probabilistic inference for future climate using an ensemble of climate model evaluations, Clim.

Change 81 3 — Fricker, J. Oakley, N. Sims, K. Signal Process. Lehoucq, D. Sorensen, C. Vernon, M. Goldstein, R. Bower, Galaxy formation: a Bayesian uncertainty analysis, Bayesian Anal. Barbarino, O. Bilgen, R. Ajaj, M. Friswell, D. Inman, A review of morphing aircraft, J. Xia, M. Friswell, E. Saavedra Flores, Equivalent models of corrugated panels, Int.

Solids Struct. Dayyani, S. Ziaei-Rad, H. Salehi, Numerical and experimental investigations on mechanical behavior of composite corrugated core, Appl. Powell, H. Loeppky, J. Welch, Choosing the sample of a computer experiment: a practical guide, Technometrics 51 — Related Papers. An efficient, higher-order perturbation approach for flow in randomly heterogeneous porous media via Karhunen-Loeve decomposition. So for instance, an author interested in curved domains might replace the triangles with curved primitives, and so might describe the elements as being curvilinear.

On the other hand, some authors replace "piecewise linear" by "piecewise quadratic" or even "piecewise polynomial". The author might then say "higher order element" instead of "higher degree polynomial". Finite element method is not restricted to triangles or tetrahedra in 3-d, or higher order simplexes in multidimensional spaces , but can be defined on quadrilateral subdomains hexahedra, prisms, or pyramids in 3-d, and so on. Higher order shapes curvilinear elements can be defined with polynomial and even non-polynomial shapes e.

More advanced implementations adaptive finite element methods utilize a method to assess the quality of the results based on error estimation theory and modify the mesh during the solution aiming to achieve approximate solution within some bounds from the exact solution of the continuum problem. Mesh adaptivity may utilize various techniques, the most popular are:. Such matrices are known as sparse matrices , and there are efficient solvers for such problems much more efficient than actually inverting the matrix.

For problems that are not too large, sparse LU decompositions and Cholesky decompositions still work well. For instance, MATLAB 's backslash operator which uses sparse LU, sparse Cholesky, and other factorization methods can be sufficient for meshes with a hundred thousand vertices. A separate consideration is the smoothness of the basis functions. For second order elliptic boundary value problems , piecewise polynomial basis function that are merely continuous suffice i. For higher order partial differential equations, one must use smoother basis functions. The example above is such a method.

## Stochastic Finite Elements: A Special Approach

If this condition is not satisfied, we obtain a nonconforming element method , an example of which is the space of piecewise linear functions over the mesh which are continuous at each edge midpoint. Typically, one has an algorithm for taking a given mesh and subdividing it. If the main method for increasing precision is to subdivide the mesh, one has an h -method h is customarily the diameter of the largest element in the mesh.

If instead of making h smaller, one increases the degree of the polynomials used in the basis function, one has a p -method. If one combines these two refinement types, one obtains an hp -method hp-FEM. In the hp-FEM, the polynomial degrees can vary from element to element. High order methods with large uniform p are called spectral finite element methods SFEM. These are not to be confused with spectral methods. The generalized finite element method GFEM uses local spaces consisting of functions, not necessarily polynomials, that reflect the available information on the unknown solution and thus ensure good local approximation.

The effectiveness of GFEM has been shown when applied to problems with domains having complicated boundaries, problems with micro-scales, and problems with boundary layers. The mixed finite element method is a type of finite element method in which extra independent variables are introduced as nodal variables during the discretization of a partial differential equation problem.

The hp-FEM combines adaptively, elements with variable size h and polynomial degree p in order to achieve exceptionally fast, exponential convergence rates. The hpk-FEM combines adaptively, elements with variable size h , polynomial degree of the local approximations p and global differentiability of the local approximations k-1 in order to achieve best convergence rates.

It extends the classical finite element method by enriching the solution space for solutions to differential equations with discontinuous functions. Extended finite element methods enrich the approximation space so that it is able to naturally reproduce the challenging feature associated with the problem of interest: the discontinuity, singularity, boundary layer, etc. It was shown that for some problems, such an embedding of the problem's feature into the approximation space can significantly improve convergence rates and accuracy.

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Moreover, treating problems with discontinuities with XFEMs suppresses the need to mesh and remesh the discontinuity surfaces, thus alleviating the computational costs and projection errors associated with conventional finite element methods, at the cost of restricting the discontinuities to mesh edges. Several research codes implement this technique to various degrees: 1. It is a semi-analytical fundamental-solutionless method which combines the advantages of both the finite element formulations and procedures, and the boundary element discretization. However, unlike the boundary element method, no fundamental differential solution is required.

## Stochastic finite elements a spectral approach - CERN Document Server

It was developed by combining meshfree methods with the finite element method. Spectral element methods combine the geometric flexibility of finite elements and the acute accuracy of spectral methods. Spectral methods are the approximate solution of weak form partial equations that are based on high-order Lagragian interpolants and used only with certain quadrature rules.

Loubignac iteration is an iterative method in finite element methods. Some types of finite element methods conforming, nonconforming, mixed finite element methods are particular cases of the gradient discretisation method GDM. Hence the convergence properties of the GDM, which are established for a series of problems linear and non linear elliptic problems, linear, nonlinear and degenerate parabolic problems , hold as well for these particular finite element methods.

Generally, FEM is the method of choice in all types of analysis in structural mechanics i. This is especially true for 'external flow' problems, like air flow around the car or airplane, or weather simulation. A variety of specializations under the umbrella of the mechanical engineering discipline such as aeronautical, biomechanical, and automotive industries commonly use integrated FEM in design and development of their products. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and also in minimizing weight, materials, and costs.

FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of both modeling and analysis of a system. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured.

The mesh is an integral part of the model and it must be controlled carefully to give the best results. Generally the higher the number of elements in a mesh, the more accurate the solution of the discretised problem. However, there is a value at which the results converge and further mesh refinement does not increase accuracy. This powerful design tool has significantly improved both the standard of engineering designs and the methodology of the design process in many industrial applications.

In the s FEA was proposed for use in stochastic modelling for numerically solving probability models [24] and later for reliability assessment. From Wikipedia, the free encyclopedia.

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Numerical method for solving physical or engineering problems. For the elements of a poset , see compact element. Navier—Stokes differential equations used to simulate airflow around an obstruction. Natural sciences Engineering. Order Operator. Relation to processes. Difference discrete analogue Stochastic Stochastic partial Delay. General topics.