We present and analyze a new hybrid stochastic finite element method for solving eigenmodes of structures with random geometry and random elastic modulus. The fundamental assumption is that the smallest eigenpair is well defined over the whole stochastic parameter space. The geometric uncertainty is resolved using collocation and random material models using Galerkin method at each collocation point. The response statistics, expectation and variance of the smallest eigenmode, are computed in numerical experiments. The hybrid approach is superior to alternatives in practical cases where the number of random parameters used to describe geometric uncertainty is much smaller than that of the material models.

In standard engineering models many physical quantities such as material parameters are taken to be constant, even though their statistical nature is well understood. Similarly, assumptions of geometric constants, such as thickness of a structure, are not realistic due to manufacturing imperfections. In a detailed report on a state-of-the-art verification and validation process comparing modern simulations with the set of experiments performed in the Oak Ridge National Laboratory in the early 70s, Szabo and Muntges report discrepancies of over 20% in some quantities of interest [

The modern era of uncertainty quantification starts with the works of Babuska et al. [

The solution methods can broadly speaking be divided into

In this paper our focus is on effects of material models and manufacturing imperfections of geometric nature. It should be noted that in the context of this paper it is assumed that the problems are positive definite and the eigenpair of interest is the ground state, that is, the one with the smallest eigenvalue which, in theory, can be a double eigenvalue. Our experimental setup is nonsymmetric and, thus, a spectral gap exists and the inverse iteration converges to the desired eigenmode.

In stochastic eigenvalue problems one must address two central issues that do not arise in stochastic source problems: first, the eigenmodes are defined only up to a sign and, second, the eigenmodes must be normalized over the whole parameter space; that is, every realization must be normalized in the same way. Here our quantity of interest is the eigenvalue and, therefore, we do not necessarily have to fix the signs. The normalization is handled by solution of a nonlinear system of equations as in [

The main result of this paper is the new hybrid algorithm which combines a nonintrusive outer loop (collocation) with an intrusive inner one (Galerkin). The randomness in geometry is resolved using collocation and that of materials with Galerkin

Two realisations of the domain with meshes. The nominal domain is the reference. The rotation and scaling of the cavity as well as the direction of the uncertainty in the Young’s modulus is indicated.

Nominal domain

Perturbed domain

The rest of the paper is organized as follows: first the abstract problem is introduced in Section

We start by presenting the deterministic system of Navier’s equations of elasticity and the corresponding weak form. We then extend this to the stochastic setting in order to cover the case of uncertain domain and modulus of elasticity. We aim to compute statistics of the smallest eigenvalue of the resulting stochastic system. The geometry of the computational domain is illustrated in Figure

We use the Navier equations of elasticity to model the system of interest. Find the eigenvalue

Let us introduce a function space

We introduce two probability spaces

The geometry of the computational domain

Suppose that for each

The stochastic extension of eigenproblem (

In most cases we are mainly interested in computing statistical moments of the solution. Suppose that we have a random variable

For purposes of numerical treatment we need to approximate the random Young modulus using only a finite number of random variables. A natural way of achieving this is to use the truncated Karhunen-Loève expansion. Once a finite-dimensional approximation has been obtained we may express our solution in a generalized polynomial chaos basis and apply solution schemes based on the stochastic Galerkin finite element method; see [

The Karhunen-Loève expansion is a representation of a random field as a linear combination of the eigenfunctions of the associated covariance operator. Truncating the resulting series we obtain a finite-dimensional approximation of the original random field. The Karhunen-Loève expansion is the optimal choice among linear expansions in the sense that it minimizes the mean square truncation error [

For a fixed

The Karhunen-Loève expansion of the random field

In numerical computations we replace the random field

It is essential for numerical algorithms that the random variables

We employ the generalized polynomial chaos framework which essentially means representing our solution on a basis of orthogonal polynomials. In our case we assume the input random variables to be uniformly distributed and the polynomial chaos basis is therefore given by tensorized Legendre polynomials. The use of orthogonal polynomials as a basis allows us to apply stochastic Galerkin based methods and ensures optimal convergence of these methods; see [

Assume that

The system

For numerical computations we truncate expansion (

The most important reason for using the polynomial chaos basis is that it allows fast computation of any expectations involved. This is because we may evaluate integrals over the stochastic domain analytically and are thus able to avoid numerical integration in high dimensions. Orthogonality of the Legendre polynomials yields

Here we give a short overview of the

In our setting we can use topologically fixed meshes with high-order elements with

In the following one way to construct a

Legendre polynomials of degree

The derivatives can similarly be computed by using the recursion

The integrated Legendre polynomials are defined for

Using these polynomials we can now define the shape functions for a quadrilateral reference element over the domain

There are four nodal shape functions:

Note that some additional book-keeping is necessary. The Legendre polynomials have the property

Since we want to use fixed mesh topologies even with perturbed domains, it is important to represent curved boundary segments accurately. The linear blending function method of Gordon and Hall [

In the general case all sides of an element can be curved, but in our case only one side is—as in Figure

We will present a hybrid method of stochastic finite elements for solving the stochastic eigenvalue problem (

Let us consider the stochastic eigenproblem (

Next we address the spatial discretization of (

In our problem of interest the geometry of the computational domain depends on the random variables

Denote by

The full tensor Lagrange interpolation operator is defined as the tensor product of the univariate operators. In our two-dimensional case this is

The expected value of a random variable

We choose to solve (

We may consider the spectral inverse iteration as a stochastic extension of the deterministic inverse iteration which has been given in Algorithm

Fix

Solve

Set

Set

Stop if

The idea in the spectral inverse iteration is to interpret each equation involved in Algorithm

For some

Let us next consider the normalization step in Algorithm

For computing the eigenvalue we need to be able to evaluate the Rayleigh quotient

Algorithm

Fix

With

Solve

Set

Stop if

We may easily calculate statistical moments of a random variable

Let us consider a problem with uncertain domain and uncertain field. The computational domain is

In Figure

Case A: effect of perturbation on eigenmodes. Contour lines of displacement.

Contour plot:

Contour plot:

Contour plot:

Contour plot:

The (constant) material parameters are Poisson ratio

We will proceed in two steps by first letting only the geometry vary before adding uncertainty to the field as well. The cases are

uncertain domain with deterministic field, solved with collocation,

uncertain domain with uncertain field, solved with the new hybrid method.

The values of the smallest eigenvalue over the parameter space are shown in Figure

Case A: eigenvalues over the parameter space: the standard deviation is

Contour plot:

Surface plot:

We consider a collocation sequence, where the collocation points are taken to span all 16 cases from

Case A: convergence in eigenvalue: relative error versus collocation sequences in log-scale. Dashed line: fixed deterministic discretization in reference. Solid line: higher resolution reference. Plotmarkers: uniform collocation grids.

Expected value

Variance

The dashed line shows the convergence in the case when the deterministic discretization of the reference solution is used also in collocation. The exponential convergence in expectation is due to convergence in parameter space only. The poorly performing grids in Figure

We proceed to consider a fixed collocation grid with

At each collocation point we employ the spectral inverse iteration given in Algorithm

Case B: convergence in eigenvalue in the case of the

Expected value

Variance

The dashed horizontal line in Figure

Algorithm

Case B: energies of the spectral components

The Monte Carlo reference results have been computed from 620000 draws using 432 single core CPU hours on Intel Xeon (2009, 2.8 GHz). The anisotropic collocation took two and the hybrid algorithm for the highest reported number of multi-indices (

Notice that in more elaborate models for geometric uncertainty with much higher dimensions of the parameter space the asymptotic convergence rates will apply to the collocation method. For lower dimensional cases it is possible to converge at faster rates, of course, but in real-life applications one should consider the modeling error as well. Similarly, for the Galerkin scheme the size of multi-index sets is too small for the energies of the spectral components to exhibit asymptotic behaviour.

We have demonstrated the practicality of the new hybrid algorithm in a practical, yet idealized, application. The combination of collocation for the geometric uncertainty and Galerkin for that of materials allows us to take advantage of the higher computational efficiency of the Galerkin approach whilst keeping mesh generation simple. Indeed, in this paper we have kept the mesh topologically fixed. The observed convergence rates for the smallest eigenvalue are in line with theoretical predictions.

The main two remaining issues for future research are the mapping of the eigenmodes to the nominal domain and handling of double eigenvalues in the general case. In 2D we are confident that this extension can be readily done, but in 3D there are still open mathematical questions on how to connect the nominal domain and the perturbed ones.

The authors declare that there is no conflict of interests regarding the publication of this paper.