1 Introduction

The scientific research and technical developments in the field of new design processes are presently driven by the requirement of time compression and the need to manage complex mechatronic systems. It has been since many years now that powerful CAE tools have been made available to the designer for the assessment of specific aspects like e.g. mechanical properties, fluid dynamic behaviour, motion simulation, etc. Single parts, mechanisms or machines were validated against their specific requirements but testing and validation of the complete system often came later, resulting in risky and expensive reworking and modifications.

The present availability of commercial tools that can be integrated in a single design framework makes possible the realization of virtual prototyping environments where the designer can simulate the whole mechatronic system from the conceptualization phase up to the physical protoyping and testing phase, with a drastic reduction of the processing times. In past years, such tools have been exploited by the researchers of the Polytechnic University of Marche in different engineering fields. As an example, the paper reports three case studies: first, a comprehensive experimental analysis was performed on timing belt transmissions by means of Laser Doppler Vibrometry for vibration measurement, Digital Image Correlation for deformation maps assessment and acoustic intensimetry for noise emission; second, a computational fluid dynamics code was integrated with a multi-body dynamics software for the study of a swimming robot; third, CAD, FEM and image generation were combined to develop a test simulator able to optimize the shape of specimens used for material characterization.

2 Current Features and Open Issues of CAE Software

In the last years, the use of Computer Aided Engineering (CAE) tools has significantly increased in both industrial and academic environments. In fact, they allow the reduction of the design costs and shorten the time to market and, furthermore, they are employed by researchers as verification, troubleshooting and analysis tools. Nowadays, the predictive capability of CAE software is so advanced that often much of the design verification is performed by means of computer simulation rather than by direct physical prototype testing, which is deferred to the very last design phases.

The dynamics of multibody systems is managed by many computer codes which differ one from the other under several points of view: model description, choice of basic principles of mechanics and topological structure [23]. The research on novel simulation tools is nowadays focusing on phenomena characterized by a challenging level of mathematical complexity. Among others, the problems of systems with non-holonomic constraints [16], systems with close to singular dynamic behaviour [24], and the effect of lubrication on mechanisms surfaces [17] have been recently investigated. Due to the growing relevance of mechatronic systems working under closed-loop control, many techniques are being tested for the development of simplified gray- or black-box models aimed at implementation in governors or controllers [8], especially in the fields of robotics.

Another topic which has been arousing a particular interest in the last years is the interaction of mechanical structures with fluids. This class of problems represents still today a real challenge for both the involved mathematical complexity and the computational burden required by numerical computations. The problem of fluid-structure interaction was firstly tackled at the beginning of the space age, when the behaviour of liquids in moving tanks was a novel topics; however the subject is far from being definitely filed and much research is still on-going [25]. For example, in the field of ultra-deep waters installations of submarine pipelines efficient simulations tools for the planning and on-line management of the deployment process are greatly needed: both the simplification of models and the numerical aspects are nowadays matter of research [18].

A particular kind of interaction between fluids and multibody systems is found in aerodynamics. The current trend is that of developing tools and models for aeroelastic phenomena. This kind of simulations received a boost in the latest years due to the greater and greater computational power presently available. The results are obviously of great impact on many fields, such as the dynamic analysis of wind turbines [20] and of incompressible flows in turbomachineries [6].

Moreover, in real-world environments multiple types of coupled physical phenomena interact: the continuously increasing computational capabilities of the simulation hardware makes now possible in many situations to relax the assumptions of systems decoupling, leading to effective multiphysics simulations. New powerful analyses can now be performed in new application fields like thermal management, MEMS, electrical motors, mechanical vibrations, etc. However, coupling individual simulations may introduce limitations on stability, accuracy, or robustness that are more severe than the limitations imposed by the individual components.

Material testing is another field where CAE systems and finite element models (FEM) have been increasingly and systematically used in the last year. The testing procedures to identify the properties of materials are rapidly evolving. One of the main reasons of this change is the availability of reliable and accurate full-field techniques which allow to obtain the deformation field on specimens of any shape during a test. Full-field measurement provides indeed a lot of information on the mechanical behaviour from a single test and inverse methods can be exploited to identify the constitutive parameters [3].

Nowadays, Digital Image Correlation (DIC) [2] represents probably the most widespread full-field technique for strain measurement, thanks to its simple set-up arrangement and relatively low-cost equipment. Among the inverse methods, the most used are the Finite Element Updating Method (FEMU) and the Virtual Fields Method (VFM). Several applications on inverse methods can be found in linear elasticity, plasticity, visco-elasticity [21], etc. For each application, however, the performance of the identification procedure is strongly influenced by the accuracy of the DIC measurement and by the shape of the specimen. Simulated experiments generated through advanced CAE tools can be used to assess the error of the testing environment [4] and optimize the specimen shape and the test set-up.

3 Use Cases at the Polytechnic University of Marche

3.1 Vibro-Acoustic Simulation of Timing Belt Transmissions

The issue of the periodic noise generated by timing belts in automotive engines is relevant, therefore designers aim to reduce the arising acoustic emission; vibrations are also related to fatigue failures, therefore they affect transmissions reliability too [1]. All these facts motivated the researchers at the Dept. Industrial Engineering and Mathematical Sciences to investigate across several years the vibroacoustic behaviour of timing belts transmissions.

The main challenges of this research at the time it was developed were related to the complexity of the numerical simulation which required the integration of several numerical tools taking into account different physical phenomena. Multiphysics simulation was at its early developments and computational power was constantly increasing, but still limited. Furthermore, knowledge of composite materials behavior was not comprehensive and accurate. Similar difficulties were present on the experimental side; some vibration tests were performed with contact sensors, adding important masses to the belts and without the possibility to measure in operative conditions, with the belt running.

The problem was approached by a continuous iteration between numerical modelling and experimental validation, realized by test benches of increasing complexity. From the numerical point of view, a timing belt is a very complex system, made up of very heterogeneous and anisotropic materials (e.g. rubber, textile fibers, metal fibers), whose numerical modeling is very difficult and computationally demanding.

From the experimental point of view, it is necessary to identify measurement instruments capable of measuring deformation and vibrations under operating conditions, i.e. on a moving belt running on a set of pulleys. Since it is not possible to place sensors on the running belt, the development of the Laser Tracking Doppler Vibrometer (TLDV) offered an interesting solution to such a problem [9]. In fact, TLDV allows to track a moving point of the belt and to measure its vibration in time. This fact, by eliminating the relative motion of the measuring beam on the surface, also eliminates the source of speckle noise which always affects LDV measurements on moving surfaces.

Image processing was also a very interesting technique applied for a complete analysis of displacement and deformation of soft materials like rubber. The study of the local behavior of the belt was focused on the single tooth and on its contact with the pulley by finite element analysis, and then on the global behavior of the belt, through the development of a lumped parameter model, whose parameters are those obtained from the local analysis. The main steps of the research are shown below.

Fig. 1
figure 1

Numerical model: a deformation maps in a 3D model of a single tooth; b contact forces in the complete arc of the pulley

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Figure 1a shows the numerical model of the single tooth. In particular, the map of displacements obtained with a simplified 2D FEM model is shown and compared to the deformation map obtained in a 3D FEM model that takes into account the complex structure of the tooth and the presence of the different layers of material and reinforcing fibers. With an even more important computational effort, it was also possible to simulate the entire arc of engagement on a single pulley by means of FEM: in this way it was possible to evaluate the trend of the contact forces along the curvilinear coordinate that follows the belt trend, Fig. 1b. For a qualitative validation of the simulation, the results are compared with the sequence of images of the tooth acquired at increasing levels of deformation.

Fig. 2
figure 2

Sequence of images of the tooth at increasing levels of deformation

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To obtain a quantitative evaluation of the displacement and deformation field of the tooth, an image analysis of the deformation of a grid of points, drawn on the tooth lateral surface, was performed, see Fig. 2. As the stress level varies, the displacement of each individual target point is evaluated by processing the image Fig. 2b to obtain the displacement map Fig. 2c, which provides insight on tooth deformation. The parameters of the tooth model [13] defined in this way can be used in a lumped model composed by a system of masses, springs and dampers, in order to represent the entire arc of the belt with a lower computational weight with respect to a full FEM of the belt. The dynamic behavior obtained by this lumped parameter model can be compared with the results measured by Laser Doppler Vibrometer. Then, starting from the operational deflection shapes it is possible to calculate the emitted sound field [14]. This continuous interaction of modeling and experimental validation provided a set of tools [7, 15] that allowed a comprehensive and detailed analysis of timing belt dynamics, which triggers design optimization aimed to management of vibroacoustic emission.

3.2 Integration of CFD and Multi-body Software to Study the Dynamics of a Swimming Robot

The attempts to design machines capable of moving like marine mammals and fish are inspired by the superior performance of biological swimmers in terms of both efficiency and manoeuvrability [22]. The possibility to replicate successfully the swimming modes evolved by fish in thousands of years depends on the understanding of the fluid mechanics principles of marine locomotion. By using computational fluid dynamics simulation techniques, engineers are trying to quantify the propulsive performance of biological thrusters, i.e. tails and fins, in terms of forces and torques. However, in order to obtain the resulting motion, the aforementioned quantities must be integrated in a multi-body model, which accounts both for the mass distribution and for the hydrodynamic effects, like added mass and viscous damping, applied to the swimmer fore body. As a matter of fact, multi-body techniques can be used to evaluate the propulsive capability under different kinematic conditions, e.g. thrusters undulating frequency, and the obtained data can be exploited to improve the vehicle design in terms of mass distribution. At the same time, the dynamic equations of the model can be coupled with trajectory planners and advanced control techniques in order to compute the effort required by the robot guidance system in order to perform a given mission.

Fig. 3
figure 3

a prototype of the swimming robot; b propulsive forces and torque decomposition

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A multi-body approach has been used by authors to model an ostraciiform swimming robot they designed and manufactured in previous works [10, 11], in order to predict its motion. The vehicle, shown in Fig. 3a, consists of a rigid cylindrical hull and a bio-inspired thruster, namely an oscillating plate shaped like a caudal fin and connected to the fore body through a revolute joint. The main novelty here is the integration of the propulsive forces and torque due to the fluid-thruster interaction in the dynamic model of the whole robot.

Fluid dynamic analysis has been performed by means of computational techniques using MIGALE, an in-house developed research code based on the Discontinous Galerkin (DG) space discretization [5, 12]. The incompressible and two-dimensional version of the DG code has been used, suitably extended to deal with a moving reference frame to account for the fin oscillation. The numerical simulations provided a complete dynamic characterization of the bio-inspired thruster as a function of the Strouhal numberFootnote 1 St.

The numerical analysis provided the range of the hydrodynamic forces and torque (surge and sway force components and yaw moment, according to Fig. 3b) within an oscillation period, as a function of the foil angular position \(\theta \). In order to characterize the propulsive performance of the bio-inspired thruster, the numerical analysis has been repeated for Strouhal numbers in the range \([0.2 \div 1.1]\), where the lowest value corresponds to a negative thrust generation and the largest to approximately a half of the maximum propulsive efficiency, as shown in Fig. 4a. In other words, the range identifies two opposite swimming conditions: minimum and maximum thrust generation, while the most efficient condition stands among them.

Fig. 4
figure 4

a thrust and efficiency as a function of Strouhal number; b effect of mass distribution on maximum surge velocity and yaw oscillation amplitude (frequency \(f=2\,\mathrm {Hz}\))

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The multi-body analysis has been performed by using MSC Adams. Since most vehicle mass is due to its hull, the robotic fish has been approximated with its rigid fore body, subject to the hydrodynamic forces due to the thruster-fluid interaction. Two separate contributions have been considered to compute the vehicle inertia moments: a steady component due to the hull, the onboard electronics, the actuators and the payload; a variable component due to the ballast rod fixed to bottom of the vehicle. Although this component has a constant mass value, it can be split in two identical parts and then fixed to the hull in two specular positions with respect to the vertical plane of symmetry, i.e. the \(y-z\) plane, of the robot. In this way it is possible to modify the moments of inertia \(I_y\) and \(I_z\) according to the ballast configuration, while the position of the center of mass remains unchanged.

Simulation results provide useful hints for an effective navigation: Fig. 4b shows the effect of the mass distribution on the swim dynamics: the dashed line represents the robot oscillation amplitude about the z axis (yaw) due to the resulting moment of the propulsive forces, an effect known in literature as recoil; the diagram shows that the amplitude is reduced by one third as the moment of inertia increases. On the other hand, the continuous line shows that the ballast configuration has a minor influence on the maximum speed reached by the vehicle.

In the end, the multi-body analysis has proved its adaptability as a framework where the off-line fluid dynamics predictions can be integrated to study the dynamic behaviour of the vehicle and to test the real-time control techniques required to manage autonomous navigation.

3.3 Application of Advanced CAE Tools to Optimize the Specimen Geometry in Inverse Identification Methods

The use of inverse methods coupled with full-field measurements to identify the mechanical properties of materials has remarkably increased in the last decades, as already illustrated in Sect. 1. Nonetheless, the error assessment and the uncertainty quantification of such techniques still represent an open issue. The reason is that the identification chain includes several error sources that interact in a highly non-linear and unpredictable way. For instance, there are experimental uncertainties, errors caused by the image acquisition and the measurement technique, numerical errors due to the minimization algorithm used in the inverse method and so on. The only possibility for having a reliable error quantification is simulating the whole chain using advanced CAE tools.

One of the first contributions in this field was provided by Rossi and Pierron [19], who highlighted the importance of using simulated experiments to assess the accuracy of an inverse method and to choose the optimal experimental set-up. A simulator has been developed to virtually reproduce an experimental test: the simulator is able to disentangle with a reasonable accuracy the different error sources that come from the identification chain, i.e. experimental errors (noise, illumination, in-plane and out-of-plane movements, specimen orientation, lens distortion, etc.), DIC settings (subset size, step size, smoothing functions, etc.), errors due to the identification procedure (choice of the first guess, non-uniqueness of the solution, etc.) [4].

Fig. 5
figure 5

Different steps in the simulation process: a definition of a parametric geometry with CAD b FE model of the simulated test c DIC analysis of synthetic images generated from FEM data to reproduce a real experiment with actual uncertainties (noise, illumination, speckle pattern, etc.)

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In this article an example is illustrated where the simulator is used to optimize the geometry of specimens used to identify the non-linear hardening behaviour of metals. The different steps of the simulation process are depicted in Fig. 5. First, a parametric CAD model of the specimen is developed as a function of the geometrical variables to be investigated (Fig. 5a). Second, for a given set of geometrical variables, a complete FE model of the experiment is generated (Fig. 5b). Third, the displacement map obtained by FEM is used to deform a colormap image of the real specimen that will be analysed with DIC afterwards (Fig. 5c). The synthetic image generation has to be performed carefully to avoid the occurrence of numerical artifacts [19]. The experimental uncertainties (camera noise, illumination, rigid body movement) are introduced using suitable routines during the image generation [4]. Finally, the parameter identification is performed applying to the synthetic images the same procedure used for actual experiments, i.e., in the present example, DIC analysis to extract the strain map and VFM to identify the non-linear hardening curve. The parameters identified from the synthetic images generated with the simulator are compared with the reference parameters introduced in the FE model in order to assess the accuracy of the identification procedure.

In the case study described here, the simulator was used to reproduce a virtual experimental set-up equipped with a CMOS camera having \(1280 \times 1024\) resolution with 8-bit sensor, mounting a lens of 50 mm. The camera was supposed to be placed at 560 mm with respect to the specimen surface. Moreover, experimental uncertainties were included in the simulated images, i.e. camera noise applying an Extreme Value distribution, out-of-plane and in-plane motions, illumination variation using a simulated light spot.

The reference hardening curve is a Swift law—\(\bar{\sigma }_{REF}=k(\varepsilon _0+\bar{\varepsilon })^N\)—that reproduces the behaviour of a steel-like material, with \(K=1000\) MPa, \(\varepsilon _0=0.02\), \(N=0.5\). The global error was calculated as the root-mean-square error (RMSE) of the reference and the identified hardening curve.

Fig. 6
figure 6

Comparison of 50 specimen geometry using the simulator, different scenarios were investigated: no error (FEM), synthetic images with no noise (Image), synthetic images with noise (Noise), synthetic images with all error sources (All)

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Figure 6 shows the RMSE for 50 different specimen geometries, evaluated introducing gradually the error sources, i.e. (i) FE results—no error, (ii) synthetic images without noise, (iii) synthetic images with noise, (iv) synthetic images with all error sources. If only the FEM model were used to compare geometries, each of them would be able to identify correctly the curve. As the different error sources are introduced in the simulator, instead, a large scatter is observed and only few geometries are able to correctly identify the hardening curve. Such geometries represents the optimal solution.

4 Conclusions

The article outlines the present technological level of CAE packages and highlights the advantages that can be exploited by researchers and professionals from the integration of different software tools. As a matter of fact, it is expected that such trend will bring even more benefits when new fields of research will bring effective results in commercial implementations: just to quote some examples, they may come from the impact of web, cloud and mobile devices, from the capturing and reuse of knowledge, from the assessment of negative knowledge, from the proper management of new and smart materials, etc. The paper showed by means of three different case studies how the researchers at the Dept. of Industrial Engineering and Mathematical Sciences of the Polytechnic University of Marche exploited such issues in the past years: they are ready to take the opportunities offered by such technological developments to face the coming challenges too.