David Moens

Recent advances in non-probabilistic approaches for non-deterministic dynamic finite element analysis

SummaryThere is a growing awareness of the impact of non-deterministic model properties on the numerical simulation of physical phenomena. These non-deterministic aspects are of great importance when there is a large amount of information to be retrieved from the numerical analysis, as for instance in a numerical reliability study or reliability based optimisation during a design process. Therefore, the non-deterministic properties form a primordial part of a trustworthy virtual prototyping environment. The implementation of such a virtual prototyping environment requires the inclusion of non-deterministic properties in the numerical finite element framework. This articel gives an overview of the emerging non-probabilistic approaches for non-deterministic numerical analysis, and compares them to the classical probabilistic methodology. Their applicability in the context in engineering design is discussed. The typical implementation strategies applied in literature are reviewed. A new concept is introduced for the calculation of envelope frequency response functions. This method is explained in detail and illustrated on a numerical example.

Fuzzy Finite Element Method for Frequency Response Function Analysis of Uncertain Structures

A concept is presented for incorporating fuzzy uncertainties in dynamic e nite element analyses of uncertain structures. The objective is twofold. The e rst goal is to clarify and extend the classical fuzzy e nite element (FFE) method as it was introduced for static analyses. The shortcomings of the classical approach are described, and an extension to a generalized approach is proposed. This generalized approach is proven to be a more realistic and therefore more reliable concept for taking uncertainty into account. The second goal is to illustrate the applicability of the method for dynamic analyses. The classical and the generalized approach are compared using an eigenvalue analysis of a simple numerical example. The FFE method is also applied to the calculation of the total envelope frequency response function (FRF) using the modal superposition principle. This method requires safe approximations of the individual mode envelope frequency response functions. For this purpose a number of safe approximate optimization strategies are introduced. The numerical example shows that useful results are obtained using this FFE approach for FRF calculations.

Numerical dynamic analysis of uncertain mechanical structures based on interval fields

This paper introduces the concept of interval fields for the dynamic analysis of uncertain mechanical structures in the context of finite element analysis. The theoretic background of the concept is explained, and it is shown how it can be applied to represent dependent uncertainty in the model definition phase and in the post-processing phase. Further, the paper concentrates on the calculation of interval fields resulting from a dynamic analysis. A procedure is introduced that enables the calculation of a joint representation of multiple output quantities of a single interval finite element problem while preserving the mutual dependence between the components of the output vector. The application of the approach is illustrated using a vibro-acoustic finite element analysis.

A fuzzy finite element analysis technique for structural static analysis based on interval fields

One of the main shortcomings of current fuzzy and interval nite element procedures is that mutual dependency between multiple uncertain model parameters cannot be included in the analysis. This limit is posed by the classical interval concept, where multi-dimensional interval quantities are generally treated as hypercubes, thus ignoring all possible dependency between vector components. For this reason, most literature on this subject focuses on one-dimensional output quantities. In order to cope with this problem, this work discusses the application of the concept of interval elds for static analysis of uncertain mechanical structures in the context of fuzzy nite element analysis. The theoretic background of the concept is explained, and it is shown how it can be applied to represent dependency between parametric uncertainties. Further, the paper concentrates on the calculation of interval elds resulting from static struc- tural analysis. A procedure that enables the calculation of a joint representation of multiple output quantities of a single interval nite element problem while preserving the mutual dependency between the components of the output vector is introduced. This procedure is based on a projection of the original problem on the space composed by the classical static deformation patterns. This paper in particular introduces a novel projection in which the space of the classical deformation patterns is augmented with deviatoric parts. This novel projection leads to a better approximation of the results without a signicant increase in computation time. Finally, a numerical case study illustrates the procedure and validates the improved accuracy of the results obtained with the novel projection technique.

On the determination of fatigue properties of Ti6Al4V produced by selective laser melting

This paper provides new insights into the fatigue properties of Selective Laser Molten components made from Ti6Al4V powder particles. The SLM-process parameters are optimized and high quality SLM-parts with a relative density of 99.7% are produced. Uniaxial fatigue experiments are then performed on notched and unnotched specimens and the endurance limits are determined using the staircase method and the theory of Dixon and Mood. The presented results indicate inferior fatigue strength in comparison with conventionally produced components from Ti6Al4V. Microstructural analysis shows that this is mainly due to the anisotropy in the microstructure and the weak grain boundaries between epitaxial grains. However, the failure mechanism is consistant leading to low statistical scatter in the fatigue data. Furthermore, the critical volume method and the critical distance theory have proven to be accurate and efficient design tools to account for the notch-effect in SLM-Ti6Al4V.

Quantification of uncertain and variable model parameters in non-deterministic analysis

A multitude of models for non-deterministic structural analysis have been developed. They are all designed to predict how non-nominal input parameter values propagate through the different phases of the calculation procedure. A literature review on a number of publications that present practical examples shows that the relation between the numerical formalism that describes the uncertain or variable quantity and the physical reality is not so clear. In almost all cases the authors (have to) make assumptions on the non-deterministic nature of the physical quantity, especially for material properties. However, the sensitivity of the structural response to material parameter changes can be very significant. The authors recommend that the numerical formalism for model parameters should be well adapted to physically observed variations.

Probabilistic and Interval Analyses Contrasted in Impact Buckling of a Clamped Column

In this study we contrast two competing methodologies for the impact buckling of a column that is clamped at both ends. The initial imperfection is postulated to be co-configurational with the fundamental mode shape of the column without the axial loading. A solution is also furnished for the case when the initial imperfection is proportional to the Filonenko-Borodich “cosinusoidal polynomial”. Probabilistic and interval analyses are conducted for each case; these are contrasted on some representative numerical data.

IDENTIFICATION OF INTERVAL FIELDS FOR SPATIAL UNCERTAINTY REPRESENTATION IN FINITE ELEMENT MODELS

In the context of integrating uncertainty and variability in Finite Element (FE) models, several advanced techniques for taking both inter(between nominally identical parts) and intra-variability (spatial variability within one part) into account have recently been introduced. In the framework of non-probabilistic variability, especially the theory of Interval Fields (IF) has been proven to show promising results. Following this approach, variability in the input parameters of the FE model is introduced as the superposition of a number of base vectors scaled by interval factors. Application of the IF concept however requires identification of these parameters. Recent work has focused on the identification of interval uncertainty for the case of inter-variability. However, to the knowledge of the authors, no such techniques for identifying interval intra-variability are present in literature. This work focuses on finding a solution to the inverse problem, where the variability on the output side of the model is known from measurement data, but the spatial uncertainty on the input parameters is unknown. This paper proposes a methodology to solve this inverse problem. The uncertain simulation space, created by propagating an interval field throughout an FE model, is modelled using its convex hull. The same concept is used to model the uncertainty in the measurement space. A metric to describe the discrepancy between these convex hulls, based on the difference of their volumes and overlap, is defined and minimised in order to identify the spatial variability on the input side of the model. Validation of the methodology is performed using simulated measurement data. It is shown that numerically exact identification of a simulated measured IF is possible following the proposed methodology.

Frequency response function analysis of structures with fuzzy modal damping parameters

The ability to include non-deterministic properties in a numerical simulation process is of great value for a design engineer. In this context, the fuzzy concept has been introduced as a tool for modeling subjective uncertainty in a numerical environment. Recently, a fuzzy finite element methodology has been developed to calculate fuzzy frequency response functions (FRF) of uncertain structures. The methodology for the interval problem at the core of this procedure is based on a new hybrid approach, which consists of a preliminary optimization step, followed by an interval arithmetic procedure. The final envelope FRFs have been proven to give a very good approximation of the actual response range of the interval problem. Initially, the fuzzy method for FRF analysis was developed for damped structures based on the proportional damping principle. This paper introduces a more general approach for the damping uncertainty based on general modal damping. In the proposed procedure, the analyst can define independent modal damping parameter intervals for each mode that is taken into account in the response function. The modal damping intervals are introduced directly into the procedure. This means that there is no extra preliminary optimization necessary. It is shown how the introduction of the modal damping intervals influences the interval arithmetical procedure that calculates the envelope response function based on the modal superposition principle. In order to validate the procedure, it is applied to a realistic model with fuzzy modal damping parameters.

Derivation of an input interval field decomposition based on expert knowledge using locally defined basis functions

Abstract. In uncertainty calculation, the inability of interval parameters to take into account mutual dependence is a major shortcoming. When parameters with a geometric perspective are involved, the construction of a model using intervals at discrete locations not only increases the problem dimensionality unnecessarily, but it also assumes no dependency whatsoever, including unrealistic parameter combinations leading to possibly very conservative results. The concept of modelling uncertainty with a geometric aspect using interval fields eliminates this problem by defining basis functions and expressing the uncertain process as a weighted sum of these functions. The definition of the functions enables the model to take into account geometrically dependent parameters, whereas the coefficients in a non-interactive interval format represent the uncertainty. This paper introduces a new type of interval field specifically tailored for geometrically oriented uncertain parameters. The field has a non-interactive interval parameter in each node of the FE mesh to keep the true dimensionality of the uncertainty intact, but it obeys a bound on the gradient of the field to account for the dependency within the field.