Sascha Duczek

Simulation Methods for Guided Wave-Based Structural Health Monitoring: A Review

This paper reviews the state-of-the-art in numerical wave propagation analysis. The main focus in that regard is on guided wave-based structural health monitoring (SHM) applications. A brief introduction to SHM and SHM-related problems is given, and various numerical methods are then discussed and assessed with respect to their capability of simulating guided wave propagation phenomena. A detailed evaluation of the following methods is compiled: (i) analytical methods, (ii) semi-analytical methods, (iii) the local interaction simulation approach (LISA), (iv) finite element methods (FEMs), and (v) miscellaneous methods such as mass–spring lattice models (MSLMs), boundary element methods (BEMs), and fictitious domain methods. In the framework of the FEM, both time and frequency domain approaches are covered, and the advantages of using high order shape functions are also examined.

Simulation of Lamb waves using the spectral cell method

Today a steadily growing interest in on-line monitoring of structures is seen. Commonly referred to as structural health monitoring (SHM), the basic idea of this technique is to decrease the maintenance costs based on a continuous flow of information concerning the state of the structure. With respect to the aeronautic industry increasing the service time of airplanes is another important goal. A popular approach to SHM is to be seen in ultrasonic guided wave based monitoring systems. Since one focus is on typical lightweight materials elastic waves seem to be a viable means to detect delimitations, cracks and material degradation. Due to the complexity of such structures efficient numerical tools are called for. Several studies have shown that linear or quadratic pure displacement finite elements are not appropriate to resolve wave propagation problems. Both the mesh density and the spatial resolution needed to control the numerical dispersion are prohibitively large. Therefore, higher order finite element methods (p-FEM, SEM) are considered by the authors. One important goal is to simulate the propagation of guided ultrasonic waves in carbon/glass fiber reinforced plastics (CFRP, GFRP) or sandwich materials. These materials are typically deployed in aeronautical and aerospace application and feature a complex micro-structure. This micro-structure, however, needs to be resolved in order to capture effects like transmission, reflection and conversion of the different wave modes. It is known that using standard discretization techniques it is almost impossible to mesh the aforementioned heterogeneous materials without accepting enormous computational costs. Therefore, the authors propose to apply the finite cell method (FCM) and extend this approach by using Lagrange shape functions evaluated at a Gauss-Lobatto-Legendre grid. The latter scheme leads to the so called spectral cell method (SCM). Here, the meshing effort is shifted towards an adaptive integration technique used to determine the cell matrices and load vectors. Hence, a rectangular Cartesian grid can be used, even for the most complex structures. The functionality of the proposed approach will be demonstrated by studying the Lamb wave propagation in a two-dimensional plate with a circular hole. The perturbation is not symmetric with respect of the middle plane in order to introduce mode conversion. In the paper, an efficient method to simulate the elastic wave propagation in heterogeneous media utilizing the finite or spectral cell method is presented in detail.

Development, Validation and Comparison of Higher Order Finite Element Approaches to Compute the Propagation of Lamb Waves Efficiently

When considering structural health monitoring (SHM) applications efficient and powerful numerical methods are required to predict the behavior of ultrasonic guided waves and to design SHM systems. The existing commercial explicit finite element analysis tools based on standard linear displacement elements quickly reach their limits when applied to ultrasonic waves in thin plates, so called Lamb waves. It is known that the required temporal and spatial resolution causes enormous computational costs. One resort to overcome this problem is the application of special finite elements utilizing higher order polynomial shape functions (p > 2). The current paper is focused on the development of such higher order finite elements and the verification of their accuracy and performance. In the paper we compare and evaluate the capabilities of spectral finite elements, p-version finite elements and isogeometric finite elements. Their advantages and disadvantages with respect to ultrasonic wave propagation problems are discussed and their properties are demonstrated by solving a benchmark problem. Higher order finite elements with varying polynomial degrees in longitudinal and transversal direction (anisotropic ansatz space) are investigated and convergence studies are performed. The results of the convergence studies are summarized and a guideline to estimate the optimal discretization is prepared. If the required accuracy is known, the proposed guideline provides a helpful means to determine both the element size and the polynomial degree template for a given model.

Anisotropic hierarchic finite elements for the simulation of piezoelectric smart structures

Purpose – Piezoelectric actuators and sensors are an invaluable part of lightweight designs for several reasons. They can either be used in noise cancellation devices as thin‐walled structures are prone to acoustic emissions, or in shape control approaches to suppress unwanted vibrations. Also in Lamb wave based health monitoring systems piezoelectric patches are applied to excite and to receive ultrasonic waves. The purpose of this paper is to develop a higher order finite element with piezoelectric capabilities in order to simulate smart structures efficiently.Design/methodology/approach – In the paper the development of a new fully three‐dimensional piezoelectric hexahedral finite element based on the p‐version of the finite element method (FEM) is presented. Hierarchic Legendre polynomials in combination with an anisotropic ansatz space are utilized to derive an electro‐mechanically coupled element. This results in a reduced numerical effort. The suitability of the proposed element is demonstrated usi...

The Finite Pore Method: a new approach to evaluate gas pores in cast parts by combining computed tomography and the finite cell method

Abstract Although pores are inevitable in casting processes, there is no secured way of evaluating their influence on the performance of cast parts, yet. The durability and dimensioning are both influenced by these discontinuities, therefore it is important to take them into account both for the design process and structural simulations. With the advent of fast and high resolution computed tomography, three-dimensional information concerning the location and the geometry of pores can be obtained. Using these data, we propose a fully automated numerical method to evaluate their influence on the mechanical properties. The current paper illustrates the general idea of how to assess such discontinuities based on voxel (combination of the words volume and pixel (picture element)) or surface tessellation language data using high order fictitious domain methods. The developed methodology provides an insight regarding the stress state of the structure and its life span. The results are used to reduce the costs in the casting process by minimising the number of discarded cast parts.

Higher Order Finite Element Methods

The efficiency of numerical methods for wave propagation analysis is essential, as very fine spatial and temporal resolutions are required in order to properly describe all the phenomena of interest, such as scattering, reflection, mode conversion, and many more. These strict demands originate from the fact that high-frequency ultrasonic guided waves are investigated. In the current chapter, we focus on the finite element method (FEM) based on higher order basis functions and demonstrate its range of applicability. Thereby, we discuss the p-FEM, the spectral element method (SEM), and the isogeometric analysis (IGA). Additionally, convergence studies demonstrate the performance of the different higher order approaches with respect to wave propagation problems. The results illustrate that higher order methods are an effective numerical tool to decrease the numerical costs and to increase the accuracy. Furthermore, we can conclude that FE-based methods are principally able to tackle all wave propagation-related problems, but they are not necessarily the most efficient choice in all situations.

The Finite Cell Method: A Higher Order Fictitious Domain Approach for Wave Propagation Analysis in Heterogeneous Structures

In this chapter a recently developed novel approach to simulate the propagation of ultrasonic guided waves in heterogeneous, especially cellular lightweight structures is presented. One of the most important drawbacks of traditional finite element-based approaches is the need for a geometry-conforming discretization. It is generally acknowledged that the mesh generation process constitutes the bottleneck in the current simulation pipeline. Therefore, different measures have been taken to at least alleviate the meshing bruden. One of these attempts is the finite cell method (FCM). It combines the advantages known from higher order finite element methods (FEM; exponential convergence rates) with those of fictitious domain methods (FDM; automatic mesh generation using Cartesian grids). In the framework of the FCM we do not rely on body-fitted discretizations and therefore shift the effort typically required for the mesh generation to the numerical integration of the system matrices which is performed by means of an adaptive Gaussian quadrature. The advantage of such a procedure in the context of wave propagation analysis is seen in the fully automated analysis process. Consequently, hardly any user input is required making the simulation very robust. Notwithstanding, it can be mathematically proven that the optimal rates of convergence are retained.

A Numerical Study on the Potential of Acoustic Metamaterials

In the present contribution we are going to investigate a special class of acoustic metamaterials, i.e. synthetic foams with spherical inclusions. This study is motivated by the need for an improved acoustical behavior of engines and vehicles which is one important criterion for the automotive industry. In this context, innovative materials offering a high damping efficiency over a wide frequency range are becoming more and more important. Since there is an innumerable selection of different absorbing materials with an equally large range of properties numerical studies are inevitable for their assessment. In the paper at hand, we look at a special class of such materials in which the influence of the inclusions on the acoustical behavior is examined in detail. To this end, we vary the size, mass density, number and position of spherical inclusions. Here, the main goal is to improve the damping properties in comparison to conventional materials which can be bought off the shelf. In that regard, the lower frequency range is of special interest to us. The results show that a random distribution of the inclusions should be favored while for the other parameters values that are centered within the investigated interval are recommended.

Fundamental Principles of the Finite Element Method

In the current chapter, we derive and discuss the fundamental principles of the finite element method (FEM). Therefore, the governing and constitutive equations are briefly recalled. By means of the method of weighted residuals (MWR), we convert the strong form of the equilibrium equations into the weak form. In the book at hand, wave-based structural health monitoring (SHM) approaches are investigated for thin-walled elastic structures that are augmented with a network of piezoelectric transducers. Consequently, the governing equations have to consider the coupled electromechanical relations. Since the fundamental idea of the FEM is the subdivision of the computational domain into small entities, the so-called finite elements, the discretized version of weak form is derived in the next step. To solve the semi-discrete equations of motion, we introduce two different time-integration methods that are used throughout the chapter. In conjunction with suitable mass lumping techniques, highly efficient numerical methods are generated. Finally, we also briefly mention different mapping concepts that are commonly used to approximate the physical geometry by means of the elemental shape functions. The basic ideas being discussed in this chapter are later also applicable to other, highly efficient numerical methods that are FE-based.

Damping Boundary Conditions for a Reduced Solution Domain Size and Effective Numerical Analysis of Heterogeneous Waveguides

In the current chapter we focus on the development of numerical methods to reduce the computational effort of finite element (FE)-based wave propagation analysis and to enable the modelling of heterogeneous cellular structures. To this end, we take two different approaches: (1) implementation of damping boundary conditions to reduce the solution domain, and (2) development of methodologies to approximately capture the heterogeneities of cellular sandwich materials. The main advantage of our approach is seen in the fact that it can be implemented in commercial FE software in a straightforward fashion. Using these approaches we can study the interaction of guided waves with heterogeneous and cellular microstructures with a significantly reduced numerical effort. By means of parametric studies we then extract important variables that influence the behavior of elastic waves in sandwich panels.