Thanida Charoensuk

Chitnarong Sirisathitkul
email: chitnarong.siri@gmail.com

Upsorn Boonyang

Innocent J Macha

Jerran Santos

David Grossin

Besim Ben-Nissan

In vitro bioactivity and stem cells attachment of three-dimensionally ordered macroporous bioactive glass incorporating iron oxides

Keywords:

Three dimensionally ordered macroporous bioactive SiO 2 CaO Na 2 O P 2 O 5 glass (3DOM BG) is synthesized by using the sol gel method. After an in vitro test in simulated body fluid (SBF), the hydroxyapatite (HAp) crystalline phase is clearly formed on its surface as confirmed by X ray diffractometry (XRD) and Raman spectroscopy. Mag netic 3DOM BG/Fe samples are synthesized by partial substitution of SiO 2 with iron oxide. Whilst the HAp layer is not confirmed, energy dispersive spectroscopy (EDS), Fourier transform infrared spectroscopy (FTIR) and XRD analysis reveal calcium phosphate layer on the surface of 3DOM BG/Fe samples after the SBF soaking. The growth of HAp like layer is slower with increasing iron oxides. The initial mechanism that thought to induce bone forma tion is reduced due to the replacement of Ca 2+ with Fe ions in the glass network. The formation of HAp like layer is modified by the sedimentation of Ca and P while the nonmagnetic 3DOM BG forms the calcium phosphate by the ionic exchange following the Hench mechanism. The adult human adipose tissue derived stem cells (hADSCs) can be closely attached and well spread on the flat plate of all 3DOM BG/Fe and 3DOM BG. Without detectable cytotoxicity possibly induced by iron oxides, the osteoblast can be grown and proliferated. In addition to these bioactivity and biocompatibility, porous structures can allow their possible use in targeted drug delivery and magnetic properties of 3DOM BG/Fe can essentially be implemented in hyperthermia therapy.

Introduction

The bioactivity of bioactive glasses has been proven by either in vivo test in physiological environment or in vitro test in simulated body fluid (SBF). The in vitro test in SBF is internationally accepted valid method to verify the bone bonding ability with the formation of hydroxyl carbonate apatite (HCA) as the bone like layer [1 3]. The refined recipe of conven tional SBF was proposed to the Technical Committee ISO/TC150 of Interna tional Organization for Standardization in 2003 by Kokubo [START_REF] Kokubo | How useful is SBF in predicting in vivo bone bioactivity?[END_REF]. Similar compositions and ionic concentrations in human blood plasma and the SBF were also presented [START_REF] Kokubo | How useful is SBF in predicting in vivo bone bioactivity?[END_REF]. The bone bonding ability of bioactive glasses is determined from the biodegradability, the dissolution rate of its ions, the formation of new bone tissues and the stimulation of osteoblasts when ex posed to physiological fluids or SBF. Particularly, the temperature of the in vitro testing for HCA formation of bioactive glasses in SBF must be compa rable to the body temperature of 37 °C. The mechanisms of bone bonding reported, especially the first 5 reaction stages generally referred to as the Hench mechanism, suggestively confirm the glass bioactivity [2 6].

Initially, the network modifications of Na + and Ca 2+ ions at the in terface of glasses are dissolved and replaced by H + from the physiolog ical fluid according to ion exchange reactions in Eqs. [START_REF] Kokubo | How useful is SBF in predicting in vivo bone bioactivity?[END_REF] and [START_REF] Hench | Bioglass: 10 milestones from concept to commerce[END_REF].

Si-O-Na þ þ H 2 O → Si-OH þ þ Na þ þ OH ð1Þ
Si-O-Ca 2þ þ H 2 O → Si-OH þ þ Ca 2þ þ OH ð2Þ
The increase in pH with increasing OH -promotes the breaking of siloxane groups (Si O Si). Si(OH) 4 is partly released and silanol groups (Si OH) are developed at the surface according to Eq. [START_REF] Hench | Chronology of bioactive glass development and clinical applications[END_REF].

Si-O-Si þ H 2 O → Si-OH þ OH-Si ð3Þ
Owing to the depletion of Na + and Ca 2+ ions on the surface, the silica rich layer then condenses and repolymerises on the glass surface as follows.

HO-

OH j Si j OH -OH þ HO- OH j Si j OH -OH → HO- OH j Si j OH -O- OH j Si j OH -OH þ H 2 O ð4Þ
The Ca 2+ and PO 4 3-groups inside of glasses then migrate to the sur face at silica rich layer. By incorporating Ca 2+ and PO 4 3ions in the body fluid, the amorphous CaO P 2 O 5 film is formed. The film growth re sults in the presence of CaO P 2 O 5 rich layer on the top surface. The amorphous CaO P 2 O 5 rich layer incorporates OH -and/or CO 3 2-anions from the surrounding solution. Finally, the HCA is biologically crystal lized and new bone tissue is formed by interactions with other biomol ecules and cells [START_REF] Hench | Bioglass: 10 milestones from concept to commerce[END_REF][START_REF] Hench | Chronology of bioactive glass development and clinical applications[END_REF].

The continuous reactions after the first 5 stages lead to the implant site for bone tissue repair. The attachment of stem cells, the proliferation and the differentiation of the cells successively occur on the surface of bioactive glasses. The bone mineralization is finally developed with a strong bond to the implant surface [START_REF] Hench | Chronology of bioactive glass development and clinical applications[END_REF][START_REF] Hench | Third-generation biomedical materials[END_REF].

The in vitro osteogenesis is normally used to assess the potential of the bioactive glasses in bone regeneration. Several human osteoblasts like cells, i.e. MG 63 cells [START_REF] Peter | Novel biodegradable chitosan-gelatin/nano-bioactive glass ceramic composite scaffolds for alveolar bone tissue engineering[END_REF], ROS17/2.8 cells [START_REF] Li | Magnetic bioactive glass ceramic in the system CaO-P2O5-SiO 2 -MgO-CaF 2 -MnO 2 -Fe 2 O 3 for hyperthermia treatment of bone tumor[END_REF], bone marrow stromal cells (BMSC) [START_REF] Wu | Mesoporous bioactive glasses: structure characteristics, drug/ growth factor delivery and bone regeneration application[END_REF], HeLa cells [START_REF] Wang | One-pot synthesis of magnetic, macro/mesoporous bioactive glasses for bone tissue engineering[END_REF], mesenchymal stromal cells (MSC) [START_REF] Santocildes-Romero | The osteogenic response of mesenchymal stromal cells to strontiumsubstituted bioactive glasses[END_REF] were studied for the cell adhesive on the surface of bioactive glass samples. These cells were cultured in the mediums (MEM, DMEM, etc.) and supplements (fetal bovine serum, penicillin, strepto mycin, etc.) with the controlled temperature at 37 °C in an incubator with simulated human body conditions [13 14]. The biocompatibility in these studies are determined by the cell adhesion ability. The attach ment and subsequent proliferation of the osteoblasts like cells have been frequently reported. The morphology and spreading of cells de pend on several factors such as the type of cells, the composition of bioactive glass samples and the surface micro and macro structure and roughness. The bridging of cells over small pores has been observed. The modifications of porous structures and scaffolds significantly enhance the attachment and proliferation of cell [START_REF] Wu | Mesoporous bioactive glasses: structure characteristics, drug/ growth factor delivery and bone regeneration application[END_REF]. The interconnected porous structure of the scaffolds supports the vascularization, nutrient as well as allow their use for targeted drug delivery and the tissue growth. By contrast, like any ceramic brittle material its porosity weakens mechan ical properties. It has been reported that whereas the doping of MnO 2 and Fe 2 O 3 results in the decrease of bioactivity, the cell adhesion and proliferation are facilitated [START_REF] Li | Magnetic bioactive glass ceramic in the system CaO-P2O5-SiO 2 -MgO-CaF 2 -MnO 2 -Fe 2 O 3 for hyperthermia treatment of bone tumor[END_REF]. Since, the hydroxyapatite (HAp) is sim ilar to the major component of human bone, its formation on sample surfaces are believed to enhance cell adhesion and signify the potential for bone tissue regenerations [START_REF] Wang | One-pot synthesis of magnetic, macro/mesoporous bioactive glasses for bone tissue engineering[END_REF][START_REF] Gerhardt | Bioactive glass and glass-ceramic scaffolds for bone tissue engineering[END_REF]. In this study, two glass structures were produced to investigate their bioactivity and biocompatibility. Major aim of the introduction of Fe ions into the structure was their possible use in hyperthermia therapy for cancer patients. The 3DOM structure were obtained in as widely ac cepted bioactive glass (3DOM BG) and magnetic bioactive glass with the addition of Fe into the structure (3DOM BG/Fe), Both glasses were produced with hierarchical interconnected macropores similar to bone tissues. The HAp formation was monitored to investigate the bioactivity using the SBF in vitro test. The stem cells attachment after culturing was examined to verify the bioactivity and biocompatibility. The surfacial changes of 3DOM BG and 3DOM BG/Fe were inspected and compared.

The induction of osteoblast on their surfaces indicated that the ma terials developed can be used to induce the new bone tissue formation. The superparamagnetic properties were obtained by adding 10 30 mol% Fe(NO 3 ) 3 ⋅9H 2 O into SiO 2 CaO Na 2 O P 2 O 5 glasses which were aimed to be used in hyperthermia therapy.

Experimental

Preparation of 3DOM BG and 3DOM BG/Fe

Based on T. Charoensuk et al. previously published procedure [START_REF] Charoensuk | Magnetic phase transitions in macro/mesoporous bioactive glass by ferric nitrate addition in sol-gel synthesis[END_REF], four samples were synthesized with varying compositions of chemicals listed in Table 1. Sample 80S/0Fe was composed of 80SiO 2 10CaO 5Na 2 O 5P 2 O 5 and used as a nonmagnetic reference. Three 3DOM BG/ Fe samples referred to as 70S/10Fe, 60S/20Fe and 50S/30Fe were re spectively prepared with Fe composition of 10, 20 and 30 mol%. Mag netic iron oxide was reduced and incorporated under the reducing carbon environment produced by the thermal decomposition of P123 during the calcination in air atmosphere [START_REF] Wang | One-pot synthesis of magnetic, macro/mesoporous bioactive glasses for bone tissue engineering[END_REF].

In vitro SBF test

The bioactivity of the obtained samples were examined using r SBF at pH 7.25 following Kokubo method [START_REF] Kokubo | How useful is SBF in predicting in vivo bone bioactivity?[END_REF]. The grainy 3DOM BG and 3DOM BG/Fe were soaked in SBF. The SBF was daily refreshed and kept at 37 °C to simulate the in vivo situation. The soaking intervals were varied up to 28 days. After soaking, the samples were removed from the SBF, washed 3 times with deionized water and then air dried at the room temperature. The formation of HCA on their surfaces was moni tored by field emission scanning electron microscopy (FESEM), energy dispersive spectroscopy (EDS), X ray diffractometry (XRD), Raman and Fourier transform infrared (FTIR) spectroscopy.

Stem cells attachment

Adult human adipose tissue derived stem cells (hADSCs) tissue cul ture was conducted under aseptic sterile conditions in a class II laminar flow hood (Clyde Apac BH2000 series). The hADSCs were cultured till sub confluence at 5 × 10 4 cells/cm 2 T25 culture flask (Nunc) in Dulbecco's modified Eagle medium (D MEM: Glutmax)/nutrient mix ture F12 (Gibco) with 10% fetal bovine serum (FBS: Invitrogen) and in cubated at 37 °C at 5% CO 2 . Subsequently, cells were passaged and diluted to obtain the seeding density of 10 4 cells/cm 2 . They were placed on the flat plate of 3DOM BG and 3DOM BG/Fe samples for 14 days for the continual growth and adherence test. Non adherent cells that can affect the culture condition were eliminated by replacing the media every 3 days for a normal cellular growth. After the adherence test, media was decanted. The cells were washed in phosphate buffered sa line (PBS) and fixed in 4% formalin. They were then washed in distilled water and finally dehydrated for FESEM inspection.

Results

Bioactivity of 3DOM BG

According to FESEM images in Fig. 1, the 3DOM BG (sample 80S/ 0Fe) was degraded and the HAp like feature was formed by the SBF soaking. Fig. 1(b) indicates deterioration due to the prior release of some soluble silica and Fig. 1(c) demonstrates the initial stage of the crystalline HCA growth. Ultimately, the flake like HCA crystals clearly cover the surface of the sample within 21 days as shown in Fig. 1(d). FESEM also reveals the distortion in the 3DOM structure by interaction between template and precursors due to the addition of Na 2 O. The phase separation occurs due to the increase in gel phase resulting in the microspheres on the surface of 3DOM structure as exemplified in Fig. 2(a). However, the biocompatibility is not affected and the dissolu ble efficacy as well as the HCA crystalline growth on the surface of these microspheres is confirmed in the FESEM images.

Fig. 3 compares elemental composition between before and after soaking in SBF for 14, 21 and 28 days. The quantities in wt% of all ele ments are averaged from over 10 different positions on the sample sur face. The Ca and Na are initially decreased after 14 days of SBF soaking but the SBF soaking for 21 28 days leads to large increases in Ca as well as P. For 28 days of SBF soaking, the Ca/P molar ratio is approxi mately 1.57. The Si is increased after the 14 days of SBF soaking and sub sequently decreased until close to zero for 28 days soaking. This EDS analysis confirms the formation of the crystalline HCA phase.

In Fig. 4, the transmittance peaks of FTIR spectra verify the character istics of bioactive glasses. At the top spectrum, the sample without soaking in SBF exhibits the peaks at 471, 802, 1095 cm -1 and the shoul der at 1095 1220 cm -1 corresponding to the vibration of Si O Si bond and the peak at 567 cm -1 correlates to the P O vibrational peak. In additions, the vibrational peak at 1630 and 3448 cm -1 assigned to O H bonds indicates the water trapped inside the sample. After the soaking in SBF, the aforementioned vibrational peaks are still observed. The P O peak at 567 cm -1 split into doublet peak at 565 cm -1 and Fig. 5. Raman spectra of sample 80S/0Fe before and after soaking in SBF for 28 days. 602 cm -1 and an additional P O peak can be found at 968 cm -1 . In additions, the vibrational peaks around 1380, 1414 and 1448 cm -1 can be assigned to the carbonate group [17 19].

In Fig. 5, the Raman scattering spectrum of 80S/0Fe sample does not exhibit any peak before soaking in SBF since the laser beam in the mea surement is only focused on the sample surface. After soaking in SBF for 28 days, the weak peaks at 430, 591 and 1070 cm -1 as well as the sharp and strong peak at 961 cm -1 distinctly confirm the growth of HAp crys talline phase. The peaks at 430 and 591 cm

Bioactivity of 3DOM BG/Fe

The FESEM images in Figs. [START_REF] Hench | Third-generation biomedical materials[END_REF] 9 reveal the surface of 3DOM BG/Fe samples during the bioactivity test. The feature changes after soaking in SBF suggest a different mechanism compared to nonmagnetic 80S/ 0Fe. A slight change in all 3DOM BG/Fe samples can be observed after soaking in SBF for 7 days. The skeletal walls become thicker after 14 days of soaking in SBF. Obviously, the HAp like feature clearly grows after soaking in SBF for 21 days and most surfaces are covered within 28 days of SBF soaking.

Fig. 10(a) shows the formation microspheres on some outer surface of 50S/30Fe sample similar to those in 80S/0Fe sample. After soaking in SBF, the slightly dissolved silica HAp like features are revealed as in Fig. 10(b). The elemental compositions in the 3DOM BG/Fe before and after the in vitro test are analyzed by EDS. The values averaged from over 10 positions on sample 70S/10Fe, 60S/20Fe and 50S/30Fe are respectively compared in Fig. 11(a), (b) and (c). By soaking the samples in SBF, Ca and Na are slightly changed after 14 days and then clearly increased after 21 and 28 days. The Ca/P molar ratio of all samples is only around 0.8 after 28 days and Fe in these 3DOM BG/Fe samples is increased after soaking in SBF.

FTIR spectra for sample 70S/10Fe in Fig. 12(a) show vibrational peaks related to the Si O Si bending at 465 and 1090 cm -1 and stretching at 804 cm -1 . The P O vibrational peak is observed at 565 cm -1 . According to FTIR spectra of high purity iron oxide nanopowders (N99.5%, US Research Nanomaterials, Inc.), the Fe O vibrational peaks for Fe 3 O 4 at 588 cm -1 , γ Fe 2 O 3 at 586 cm -1 and α Fe 2 O 3 at 568 cm -1 are close to the P O peak. Moreover, the Fe O vibrational peaks of Fe 3 O 4 at 1047 and 1094 cm -1 as well as γ Fe 2 O 3 at 1063 cm -1 are comparable to the Si O Si peak. Hence the Fe O peaks cannot clearly be identified. Unlike sample 80S/0Fe, the splitting of P O vibrational peak cannot be observed in sample 70S/10Fe after the SBF soaking. The vibrational peak assigned to the carbonate group is observed at 1385 cm -1 . In ad ditions, the vibrational peak related to H 2 O is located at 1632 and 3450 cm -1 .

In Fig. 12(b) and (c), samples 60S/20Fe and 50S/30Fe exhibit vibra tional peaks of Si O Si, and P O similar to those of sample 70S/10Fe. The Fe O peak can also be clearly observed at 698 cm -1 . The vibrational peak at around 1730 cm -1 corresponded to a carbonyl group and the peaks assigned to the carbonate group at 1385, 1450 and 1493 cm -1 are explicitly found with the increase in Fe composition. After soaking in SBF, only peak assigned to carbonate at 1385 cm -1 remains while other peaks correlated to carbonyl, carbonate and iron oxide gradually disappear. The splitting of P O peak is not observed. Most vibrational peaks of 3DOM BG/Fe samples tend to shift to lower wavenumbers than those of 3DOM BG (sample 80S/0Fe). This trend also corresponds to the increase in iron oxide concentration.

Before soaking in SBF, four Raman peaks can be observed in Fig. 13(a) for sample 70S/10Fe. The peak at 339 and 702 cm -1 can be assigned to γ Fe 2 O 3 [START_REF] Oh | Characterization of iron oxides commonly formed as corrosion products on steel[END_REF]. The peak at 965 cm -1 is a characteristic of the symmetric stretching of P O bond. The broad peak at 1380 cm -1 can also be attributed to the symmetric stretch of the P=O bond [START_REF] Kiani | Structural characterization and physical properties of P 2 O 5 -CaO-Na 2 O-TiO 2 glasses by Fourier transform infrared, Raman and solid-state magic angle spinning nuclear magnetic resonance spectroscopies[END_REF]. After soaking in SBF for 28 days, several Raman shifts are observed. For iron oxide peaks, the peaks at 228, 293 and 427 cm -1 are attributed to α Fe 2 O 3 and the peaks at 337, 487, 572 and 671 cm -1 correspond to γ Fe 2 O 3 [START_REF] Oh | Characterization of iron oxides commonly formed as corrosion products on steel[END_REF][START_REF] Hanesch | Raman spectroscopy of iron oxides and (oxy)hydroxides at low laser power and possible applications in environmental magnetic studies[END_REF][START_REF] Soler | Raman spectroscopy of iron oxide nanoparticles[END_REF]. However, there is a peak shift in the symmetric stretching of P O bond from 965 cm -1 to 971 cm -1 .

In Fig. 13(b) and (c), the Raman spectra exhibit comparable peaks to those of sample 70S/10Fe. For sample 60S/20Fe before SBF soaking, the peaks at 473 and 691 cm -1 correspond to γ Fe 2 O 3 and the peak at 973 cm -1 is attributed to the P O bond observed. After soaking in SBF for 28 days, the peaks corresponding to γ Fe 2 O 3 are found at 330, 487 and 719 cm -1 . The peak attributed to the P O bond is shifted to 979 cm -1 . For sample 50S/30Fe, the peaks at 334, 479 and 704 cm -1 at tributed to γ Fe 2 O 3 can be observed before SBF soaking. The peaks at 978 and 1380 cm -1 are respectively due to P O and P=O bonds. After soaking for 28 days, all peaks are still present with slightly different intensity. However, the position of P O peak is not shifted for this sample.

The XRD is used to probe the formation of HAp crystalline phase on sample surface. In Fig. 14, the XRD pattern of each 3DOM BG/Fe sample has a broad reflection in the 15 30°regime indicating the amorphous structure of the glass. Sample 70S/10Fe reveals the weakly additional

Stem cell attachment

From FESEM images in Fig. 15, the hADSCs are well attached and spread on surface of each sample after the cell culture for 14 days. These cells are over 100 μm in size and mostly triangular in shape.

Discussion

Bioactivity of 3DOM BG

The bioactivity of 3DOM BG is clearly demonstrated in the in vitro testing at the controlled temperature of 37 °C. When sample 80S/0Fe is soaked in SBF, the first reaction is the ionic exchange between Ca 2+ and Na + linked nonbridging oxygen (NBO) (Si O Ca and Si O Na) with H + from solution. The presence of Na + in this sample facilitates such exchange. This reaction results in the increase in pH due to the increase in OH -in the solution. The bridging oxygen (BO) breaking of Si O Si bonds is consequently induced by the increase in OH -and leads to the release of some silica in form of Si(OH) 4 and the formation of Si OH groups at the interface. The SiO 2 rich layer then condenses and repolymerizes on the surface depleted in Ca 2+ and Na + . Subsequently, the amorphous CaO P 2 O 5 rich film is formed on the top of the SiO 2 rich layer resulting from the Ca 2+ and PO In additions, the enhanced reactive surface areas of the 3DOM structure also support the dissolution and resorption abilities resulting in the con version to the bone like layer [START_REF] Baino | Bioactive glasses: special applications outside the skeletal system[END_REF][START_REF] Choi | Three-dimensional scaffolds for tissue engineering: the importance of uniformity in pore size and structure[END_REF][START_REF] Zhang | Cytotoxicity of three-dimensionally ordered macroporous sol-gel bioactive glass (3DOM-BG)[END_REF].

Although the absolute values are not precisely measured by EDS (it is known to be semi quantative), the relative amount of each element can be accurately analyzed (as long as standards used) by averaging values over ten different positions on the sample surface. The initial de crease of Ca and Na after SBF soaking indicates the ionic exchange reaction after the soaking. The large increases in Ca as well as P corre spond to CaO P 2 O 5 rich films formed at the top of surface and the growth of crystalline HCA. The increase of Si after the 14 days of SBF soaking may involve the formation of SiO 2 rich layer. The subsequent decrease implies the formation of CaO P 2 O 5 rich films on the surface. The minimum Si in the case of 28 days of SBF soaking indicates the fully covered surface by CaO P 2 O 5 rich films and/or HCA. After 28 days of SBF soaking, the Ca/P molar ratio of samples is similar to the 1.0 2.0 range of the value present in human body [START_REF] El-Meliegy | Glasses and Glass Ceramics for Medical Applications[END_REF][START_REF] Paiva | In vitro studies of bioactive glass/polyhydroxybutyrate composites[END_REF]. The characteristic bonds of glass are revealed with the Si O Si and P O vibrational peaks in FTIR spectra. The presence of amorphous phosphate groups confirms the HAp formation after the in vitro SBF test. The vibrational peaks assigned to carbonate group also indicate the formation of crys tallized HCA [START_REF] Adams | Sol-gel synthesis of SiO 2 -CaO-Na 2 O-P 2 O 5 bioactive glass ceramic from sodium metasilicate[END_REF][START_REF] Lutisanova | Comparison of bioactivity in vitro of glass and glass ceramic materials during soaking in SBF and DMEM medium[END_REF]. In additions, the Raman shift and XRD spectra clearly confirm the bioactivity of 3DOM BG.

Bioactivity of 3DOM BG/Fe

FESEM images demonstrate the HAp formation is apparently re duced with the increase in Fe concentration. The first stage of Hench mechanism involving the ion exchange may be disrupted. The Fe is in corporated in 3DOM BG/Fe samples at the expense of Ca [START_REF] Wu | Multifunctional magnetic mesoporous bioactive glass scaffolds with a hierarchical pore structure[END_REF]. The release of Ca 2+ from 3DOM BG/Fe surface which induces the silica dis solution is decreased. However, the feature resembled HAp crystalline phase is formed on the surface by the agglomeration of Ca and P from the SBF. Although the microspheres disrupt some part of 3DOM struc ture, they do not affect the bioactivity of 3DOM BG/Fe samples.

After the SBF soaking, the changes in all elements observed by EDS can be explained according to the Hench mechanism. Initially, the de creases of Ca and Na signify the first stage of ion exchange. The slight changes of Ca and Na correspond to feature changes previously shown by the FESEM images. Subsequent increases in Si as well as Ca and P are due to the formation of SiO 2 rich layer and CaO P 2 O 5 rich film, re spectively. However, the increases in Ca and P are smaller than that of the nonmagnetic 3DOM BG indicating that the presence of Fe disrupts the mechanism of HAp growth on these bioactive glasses. The Ca/P molar ratios are not in range of the value in human body. The presence of Fe reduces the ionic exchange, and hence, a less amount of silica is dissolved. The increases in Ca and P after soaking in SBF for 21 and 28 days are due to the sedimentation or agglomeration of Ca and P on the surface of samples during the process of HAp formation. However, the rate is slow since the Ca/P ratio does not reach the value in human body after 28 days of SBF soaking. So, all 3DOM BG/Fe samples are de cidedly less bioactive than the nonmagnetic 3DOM BG. The increase of Fe after soaking in SBF indicates the minimal release from the surface. This finding suggests that iron oxides will be preserved within the struc ture when these magnetic bioactive glasses are implemented in hyper thermia treatment.

The Fe O bonds probably overlap the FTIR vibrational peaks with characteristic peaks of bioactive glass [START_REF] Wang | One-pot synthesis of magnetic, macro/mesoporous bioactive glasses for bone tissue engineering[END_REF][START_REF] Liu | Magnetic 3-D ordered macroporous silica templated from binary colloidal crystals and its application for effective removal of microcystin[END_REF][START_REF] Tartaj | The preparation of magnetic nanoparticles for applications in biomedicine[END_REF]. After soaking in SBF, the changes in structural bonding occurred and the effect of iron oxide concentration result in the lower wavenumbers of most vibrational peaks. In additions, Raman spectra clearly show the bioactivity of 3DOM BG/Fe. The peak shift corresponding to the symmetric stretching of P O bond is clearly correlated to the increase in Fe concentration. These peak shifts indicate the increase in average length of P O bonds [START_REF] Kiani | Structural characterization and physical properties of P 2 O 5 -CaO-Na 2 O-TiO 2 glasses by Fourier transform infrared, Raman and solid-state magic angle spinning nuclear magnetic resonance spectroscopies[END_REF]. The peak shift of P O bond, also influenced by the SBF soaking, is decreased by the increase in Fe composition. For the 50S/30Fe sample with the highest concentration of iron oxide, the peak shift due to the SBF soaking does not occur like in other cases. Moreover, the Raman shift is found only for the peak related to γ Fe 2 O 3 . The γ Fe 2 O 3 nanocrystals is increasingly formed with increasing amounts of Fe.

Although the in vitro bioactivity of 3DOM BG/Fe samples is evident, the HAp phase is not confirmed. The presence of Fe in bioactive glass matrix apparently disrupts the formation of HAp. Nevertheless, the Ca P layers are grown on 3DOM BG/Fe surfaces as shown in FESEM im ages and XRD patterns.

Stem cell attachment

The attachment and well spread of cells on all sample surfaces indi cating the good biocompatibility. While the degradability and the HAp crystalline phase formation are important to initiate the bone tissue re generation, the low biodegradability of 3DOM BG/Fe does not disrupt the cell adhesion. The presence of Fe composition reveals no cytotoxic ity. The osteoblast is an indication of the capacity of the growth and pro liferation on the 3DOM BG as well as 3DOM BG/Fe samples synthesized in this research.

Conclusions

In this study, the initial step of HAp growth according to the Hench mechanism leading to the bone bonding on 3DOM BG is clearly shown. The XRD, FTIR and Raman spectra confirm the structure of chemical bonds and functional groups of bioactive glasses corresponding to the HAp formation during the SBF soaking. The in vitro test by soaking 3DOM BG/Fe samples in SBF indicates different bioactivity from that of the 3DOM BG. The HAp like layer has a lower growth rate in the case of higher Fe concentration. The EDS indicates that the Ca/P ratio is not close to the value found in human bone even after soaking in SBF for 28 days. However, the well adhesion and spread of hADSCs on the flat plate sample after 14 days of the stem cell culture also confirm the bio compatibility of both 3DOM BG and 3DOM BG/Fe. The inclusion of iron oxide does not lead to cytotoxicity and the osteoblast can be grown and proliferated. The presence of iron oxide with superparamagnetic be havior is useful for hyperthermia therapy owing to the induced heat under the alternating magnetic field. The magnetization is increased with the increase in Fe(NO 3 ) 3 •9H 2 O from 10 to 30 mol%. The potential applications of these multifunctional bioactive glasses us to replace, re pair and regenerate bone defects especially in the bone cancer therapy.

Fig. 1 .

 1 Fig. 1. FESEM images of 80S/0Fe 3DOM-BG (a) before soaking in SBF and after soaking in SBF for (b) 7 days, (c) 14 days, (d) 21 days, (e) 28 days and (f) 28 days shown in a lower magnification.

Fig. 3 .

 3 Fig.3. EDS analysis of elemental composition of 80S/0Fe sample before and after soaking in SBF.

Fig. 2 .

 2 Fig. 2. FESEM images of sample 80S/0Fe showing (a) silica dissolution from microspheres after soaking in SBF for 7 days and (b) HAp crystalline growth on the surface of microsphere after soaking in SBF for 28 days.

Fig. 4 .

 4 Fig. 4. FTIR spectra of sample 80S/0Fe before and after soaking in SBF.

  -1 are related to the O P O bending mode. The peaks at 961 and 1070 cm -1 correspond to the sym metric stretching mode of P O bond of PO 4 3and the asymmetric stretching mode of P O bond [20 21]. Likewise, the XRD patterns shown in Fig. 6 indicate the characteristics of bioactive glasses. A broad reflection around 15 30°indicates with amorphous nature of this glassy sample. Before soaking in SBF, additional reflections at 31.7, 34.5 and 48.4°respectively corresponding to (330), (331) and (541) planes are assigned to calcium phosphate silicate (JCPDS 040 0393). After soaking in SBF for 14 days, the peak positions at 26.0, 32.0, 39.8, 46.9, 49.8 and 53.3°are assigned to (002), (211), (310), (222), (213) and (004) reflections of HAp, respectively (JCPDS 09 0432).

Fig. 7 .

 7 Fig. 7. FESEM images of sample 70S/10Fe (a) before soaking in SBF and after soaking in SBF for (b) 7 days, (c) 14 days, (d) 21 days, (e) 28 days and (f) 28 days shown in a lower magnification.

Fig. 6 .

 6 Fig.6. XRD spectra of sample 80S/0Fe before and after soaking in SBF for 14 days.

Fig. 8 .

 8 Fig. 8. FESEM images of sample 60S/20Fe (a) before soaking in SBF and after soaking in SBF for (b) 7 days, (c) 14 days, (d) 21 days, (e) 28 days and (f) 28 days shown in a lower magnification.

Fig. 9 .Fig. 10 .

 910 Fig. 9. FESEM images of sample 50S/30Fe (a) before soaking in SBF and after soaking in SBF for (b) 7 days, (c) 14 days, (d) 21 days, (e) 28 days and (f) 28 days shown in a lower magnification.

Fig. 11 .

 11 Fig. 11. Comparison of EDS composition analysis of 3DOM-BG/Fe samples before and after soaking in SBF for 14, 21 and 28 days; samples (a) 70S/10Fe, (b) 60S/20Fe and (c) 50S/ 30Fe.

Fig. 12 .

 12 Fig. 12. FTIR spectra of 3DOM-BG/Fe samples before and after soaking in SBF for 6 h-14 days; samples (a) 70S/10Fe, (b) 60S/20Fe and (c) 50S/30Fe.

4 3 -

 3 migration to the top. This film is incorporated by Ca 2+ and PO 4 3from solution leading to the film growth. It follows that the crystalline HCA is formed by the incorpora tion of OH -and CO 3 2-from the solution. These phenomena involving the first 5 stages of the bone bonding reaction in the Hench mechanism [2,6] are explicitly revealed by FESEM images. Conclusively, the Na 2 O additive promotes the degradability and bioactivity of the 3DOM BG.

Fig. 14 .

 14 Fig. 14. XRD spectra of 3DOM-BG/Fe samples of (a) 70S/10Fe, (b) 60S/20Fe, and (c) 50S/ 30Fe before and after soaking in SBF for 14 days.

Fig. 13 .

 13 Fig. 13. Raman spectra of 3DOM-BG/Fe samples before and after soaking in SBF for 28 days; samples (a) 70S/10Fe, (b) 60S/20Fe and (c) 50S/30Fe.

Fig. 15 .

 15 Fig. 15. FESEM images of cells attachment on surface of samples (a) 80S/0Fe, (b) 70S/10Fe, (c) 60S/20Fe and (d) 50S/30Fe after culturing for up to 14 days.

Table 1

 1 Chemicals used in the synthesis of 3DOM-BG and 3DOM-BG/Fe.

	Sample	Composition SiO 2 -CaO-Na 2 O-P 2 O 5 -Fe	P123	TEOS	TEP	1 M NaNO 3	Ca(NO 3 ) 2 •	Fe(NO 3 ) 3 •9H 2 O	1 M HNO 3	EtOH
	codes	(mol%)	(g)	(mL)	(mL)	(mL)	4H 2 O (g)	(g)	(mL)	(mL)
	80S/0Fe	80-10-5-5-0	0.667	1.79	0.15	1.00	0.236	-	1.00	2.50
	70S/10Fe	70-10-5-5-10	0.581	1.57	0.15	1.00	0.236	0.404	1.00	2.50
	60S/20Fe	60-10-5-5-20	0.506	1.34	0.15	1.00	0.236	0.808	1.00	2.50
	50S/30Fe	50-10-5-5-30	0.417	1.12	0.15	1.00	0.236	1.212	1.00	2.50

Acknowledgement This work is financially supported by Walailak University (Molecular Technology Research Unit grant) and Shell Centennial Education Fund (2013), Shell Companies in Thailand. Dr. D. Grossin, Dr. I. Macha and Prof B. Ben Nissan were supported by the Australian Academy of Sci ence and the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska Curie grant agreement No 645749.


F Errico

M Ichchou

F Franco

S De Rosa

O Bareille

C Droz

Schemes for the sound transmission of flat, curved and axisymmetric structures excited by aerodynamic and acoustic sources

Keywords: Sound Transmission, Turbulent Boundary Layer, Periodic Structures, Curved Structures, Multi-Layered Structures

The paper describes a methodology for computing the sound transmission loss of any flat, curved and cylindrical, homogeneous and periodic structure, under any type of acoustic and/or aerodynamic load. An approximate excitation model is introduced to reproduce uncorrelated and spatially-correlated loads using a wavenumber integration of surface waves. Then, a wave finite element formulation is developed and interfaced with the excitation models in order to cover industrially-relevant case studies. Analytical, numerical and experimental transmission losses are presented for validation purposes. Finite size effects are also taken into account using a spatial windowing and a cylindrical analogy, for curved structures. Different periodic-cell designs are also compared and investigated under turbulent boundary layer and diffuse acoustic field excitations.

Introduction

Sandwich composite structures are extensively used in modern aerospace industry as well as in the automotive, naval and civil ones because they are lighter and stronger than most advanced panels in aluminium alloys. The anisotropy of such structures can be easily modified by changing the material and the shape of the core, obtaining different wave propagation properties. However, these types of structures are also known for having poor vibroacoustic performances which, often, can result in higher interior noise levels. This problem has a strong impact in many engineering areas, from space launchers to aircraft fuselages. Strong efforts have been recently placed on advanced methodologies for the the modelling of acoustic radiation of laminates and sandwich panels, since, classical models, using for example the finite element method (FEM), lead to cumbersome computational cost. Some FEM applications for the vibroacoustic analysis of simple structures, under random aeroacoustic loads, are present in literature [START_REF] Franco | Numerical approximations on the predictive responses of plates under stochastic and convective loads[END_REF][START_REF] Ichchou | Equivalent rain on the roof loads for random spatially correlated excitations in the mid frequency range[END_REF][START_REF] Hambric | Vibrations of plates with clamped and free edges excited by low-speed turbulent boundary layer flow[END_REF][START_REF] Rosa | A scaling procedure for the response of an isolated system with high modal overlap factor[END_REF][START_REF] Rosa | Exact and numerical responses of a plate under a turbulent boundary layer excitation[END_REF]].

An efficient alternative in terms of computational cost is, for example, the transfer matrix method (TMM). It is a general method used for the prediction of the propagation of monochromatic plane waves in planar and multi-layered structures of infinite extent [START_REF] Allard | Propagation of sound in porous media: Modelling sound absorbing materials[END_REF]. Many applications of the TMM to the modelling of sound trasmission of composite structures have already been validated [START_REF] Atalla | Modelling the sound transmission through complex structures with attached noise control materials[END_REF][START_REF] Parrinello | Transfer matrix representation for periodic planar media[END_REF].

Finite size effects, important at low frequencies, can be included through appropriate corrections, leading to a broadband accuracy of the method [START_REF] Rhazi | A simple method to account for finite size effects in the transfer matrix method[END_REF][START_REF] Leppington | The acoustic radiation efficiency from rectangular plates[END_REF][START_REF] Villot | Predicting the acoustical radiation of finite size multilayered structures by applying spatial windowing on infinite structures[END_REF].

Alternatively, the wave finite element method (WFE), specifically for homogeneous and periodic structures, allows the modelling of just a single repetitive cell, applying on it the periodicity conditions for a correct description of the entire (infinite) waveguide [START_REF] Brillouin | Wave propagation in periodic structures: Electric filters and crystal lattices[END_REF][START_REF] Mead | Wave propagation in continuous periodic structures: research contributions from southampton[END_REF][START_REF] Manconi | Modelling wave propagation in two dimensional structures using finite element analysis[END_REF][START_REF] Errico | The modelling of the flow-induced vibrations of periodic flat and axial-symmetric structures with a wave-based method[END_REF][START_REF] Renno | Calculating the forced response of cylinders and cylindrical shells using the wave and finite element method[END_REF][START_REF] Renno | Vibration modelling of structural networks using a hybrid finite element/wave and finite element approach[END_REF]. The use of finite elements, for the cell description, enhances this method allowing the description of any type of complex structural shape, even in case of curvature [START_REF] Morsbol | Elastic wave propagation in curved flexible pipes[END_REF][START_REF] Manconi | Wave characterization of cylindrical and curved panels using a finite element method[END_REF][START_REF] Manconi | The loss-factor of pre-stressed laminated curved panels and cylinders using a wave and finite element method[END_REF][START_REF] Manconi | Estimation of the loss factor of viscoelastic laminated panels from finite element analysis[END_REF]. While mainly used for the analysis of the elastic waves' propagation in periodic media, the application of the WFEM for the sound transmission of sandwich panels has been recently proposed, under a plane wave load [START_REF] Christen | Wave finite element formulation of the acoustic transmission through complex infinite plates[END_REF][START_REF] Yang | Prediction of sound transmission through, and radiation from, panels using a wave and finite element method[END_REF] or diffuse acoustic load [START_REF] Orrenius | Wave modelling in predictive vibro-acoustics: Applications to rail vehicles and aircraft[END_REF][START_REF] Orrenius | Analysis of sound transmission through aircraft fuselages excited by turbulent boundary layer or diffuse acoustic pressure fields[END_REF], even though, to authors' knowledge, no application is available for curved and complex configurations, under general loads.

Within the frameworks of curved structures, alternative methods have also been presented. A mathematical model, for the transmission of airborne noise through the walls of an orthotropic cylindrical shell, has been firstly proposed by Koval, [START_REF] Koval | On sound transmission into an orthotropic shell[END_REF][START_REF] Koval | Sound transmission into a laminated composite cylindrical shell[END_REF][START_REF] Koval | On sound transmission into a thin cylindrical shell under flight conditions[END_REF]. For curved composite laminates, the vibroacoustic problem has been further developed through a spectral approach based on a discrete lamina description, [START_REF] Ghinet | The transmission loss of curved laminates and sandwich composite panels[END_REF][START_REF] Ghinet | Diffuse field transmission into infinite sandwich composite and laminate composite cylinders[END_REF]. Other semi-analytic approaches, based on a receptance method, have also been proposed in order to analyse the sound transmission of aircraft panels with stringers and ring frames, [START_REF] Liu | Sound transmission through curved aircraft panels with stringer and ring frame attachments[END_REF][START_REF] Liu | Noise radiation of aircraft panels subjected to boundary layer pressure fluctuations[END_REF].

Periodic structures and innovative material configurations (often indicated as meta-materials), on the other hand, can be used as frequency-selecting structures. The related waveguides, because of their complex shapes, require a higher computational cost for an accurate numerical simulation.

In addition, the knowledge and the modelling of the correct operating conditions are fundamental in automotive and aerospace applications. For example, whenever a convective flow is present, boundary layer models should be included for completeness. None of the models in literature, at this stage, allows the analysis of complex structural periodic shapes, in presence of curvature and under aeroacoustic excitations, at the same time. For example, in the work of Yang et al. [START_REF] Yang | Prediction of sound transmission through, and radiation from, panels using a wave and finite element method[END_REF] only infinite flat homogenised structures can be analysed under plane waves' excitation. On the other hand, in the case of curved structures, the work of Kingan et al. [START_REF] Kingan | Application of the wave and finite element method to calculate sound transmission through cylindrical structures[END_REF] is limited to a single plane wave excitation, once a circumferential number is fixed; thus the sound transmission of complex curved structures under stochastic excitation can not be obtained, in this case.

The novelty of the present paper stands in overcoming some of these limits proposing a methodology for dealing with a wider range of case-studies, under operational conditions: space launcher fairings, fuselage panels, pipes, ducts and acoustic barriers. Periodic flat, curved and cylindrical structural designs can be compared in terms of their vibroacoustic performance, under any desired convective and acoustic load.

The paper is structured as follows: Section 2 presents a theoretical background of the WFE method; Section 3 describes the proposed load approximation and the wavenumber integration, to take into account a general fluid excitation; Sections 4 and 5 are devoted to the validations for both the acoustic and aerodynamic load; Section 6 presents an analysis of the proposed approximations and a discussion the advantages and limits of the approach. It is finally shown a comparison for the effective transmission loss of periodic cylinders as enabled by the presented method.

The Wave Finite element Method

A 2D periodic structure is composed by an assembly, along arbitrary directions, of identical elementary cells. Using any FE commercial code, the mass and stiffness matrices of the cell, whenever complex, can be extracted and post-processed. Of course, since being FEs, all classic meshing considerations for an appropriate wavelength description are valid. With reference to Fig. 1, the dynamic stiffness equation of the segment can be written as

[K -ω 2 M]q = Dq = f + e (1)
where q, f and e are respectively the nodal vectors of degrees of freedom (DoFs), internal and external forces; K, M and D are the stiffness, mass and dynamic stiffness matrices. Damping can be modelled by including, in Eq. ( 1), appropriate complex matrices and/or coefficients. In this paper, it is assumed that the vectors and matrices are ordered following the same sequence of the DoFs nodal vector: q = [q I , q F , q L , q 1 , q A , q R , q 2 , q 3 , q 4 ]. The wave motion through a periodic media can be analysed by imposing the Bloch-Floquet conditions [START_REF] Brillouin | Wave propagation in periodic structures: Electric filters and crystal lattices[END_REF][START_REF] Mead | Wave propagation in continuous periodic structures: research contributions from southampton[END_REF] to the finite element (FE) of a periodic cell, assuming time and space harmonic excitation. The periodicity conditions are translated in a magnitude and phase link among each point belonging to the periodic pattern, using a complex propagating constant for each wave type.

Displacements and forces at any point of the cell can be connected to the ones of a limited number of them, as follows:

           q A q R q 2 q 3 q 4            =            0 Iλ Y 0 0 0 0 Iλ X 0 0 0 0 Iλ X 0 0 0 Iλ Y 0 0 0 Iλ X λ Y                    q I q F q L q 1         (2)
with

λ X = e -ik X L X λ Y = e -ik Y L Y (3)
where k X and k Y are wavenumbers of the propagating (or forcing) wave in the periodicity directions X and Y , while L X and L Y represent the cell lengths along the same directions. The matrix I is, instead, the identity matrix. In a general form, thus, the total displacements and forces are connected to the reduced vectors (superscript red) through a periodicity matrix Λ (see Eq. ( 2)):

q = Λq red f = -Λf red e = -Λe red (4)
Pre-multiplying Eq. ( 1) by Λ H , where H stands for the hermitian operator, the dynamic stiffness matrix of the reduced model is given by Eq. ( 5):

D S = Λ H [K -ω 2 M]Λ. (5)
The linearity of the stress tensor with respect to the displacement field, in addition to the periodicity relations, leads to an equilibrium of the internal forces between neighbouring cells; thus only external forces are considered.

At this stage, different eigenvalue problems can be solved, if the target is the estimation of the dispersion curves of the periodic structure, [START_REF] Manconi | Modelling wave propagation in two dimensional structures using finite element analysis[END_REF][START_REF] Droz | A hybrid wave-mode formulation for the vibroacoustic analysis of 2d periodic structures[END_REF].

Modal Order Reduction: A Craig-Bampton Scheme

The use of modal reduction is highly suggested for very fine meshes. In these cases, the internal degrees of freedom, defined before as q I , are substituted by the modal participation factors [START_REF] Droz | A reduced formulation for the free-wave propagation analysis in composite structures[END_REF][START_REF] Zhou | Wave finite element method based on reduced model for one-dimensional periodic structures[END_REF].

Here an example of component mode synthesis (CMS) procedure, performed at the cell's scale, is

shown. The aim of the CMS procedure is to achieve a significant reduction of the number of inner DOFs, by replacing displacements with the local modes of the cell. Here, the displacement vector q defined in Eq. ( 1), is partitioned into the inner displacements, q In , and boundary displacements, q B . In this specific case, since the nodes belonging to the top and bottom of the cell are used for load translation, as shown in subsection 2.3, q In is a subset of the q I in Eq. [START_REF] Ichchou | Equivalent rain on the roof loads for random spatially correlated excitations in the mid frequency range[END_REF]. By using this division, Eq. (1) takes the form of Eq. ( 6):

  K BB K BIn K InB K InIn   -ω 2   M BB M BIn M InB M InIn     q B q In   =   f B 0   , (6)
where f In is zero, since no load is applied on this subset of nodes. The reduced basis involves the static boundary modes Ψ B and component modes Ψ C .

In this way the final displacements vector can be re-written as:

  q B q In   = G   q B P In   ; G =   I 0 Ψ B Ψ C   (7)
where I and 0 are the identity and zero matrix respectively, and P In is the set of retained modal participation factors. The static boundary modes Ψ B and component modes Ψ C can be derived from Eq. ( 8),

Ψ B = K -1 InIn K InB ; (K InIn -ω 2 M InIn )Ψ C = 0. (8)
In the Craig-Bampton (CB) approach, the modal selection is based on the lower resonance frequencies. This method has been extended in a wave approach context, when the aim is to capture the local deformed shape of the periodic unit cell. This means that the displacements inside a unit-cell can be expanded on a subset of stationary modes [START_REF] Droz | A reduced formulation for the free-wave propagation analysis in composite structures[END_REF][START_REF] Zhou | Wave finite element method based on reduced model for one-dimensional periodic structures[END_REF].

Finally the stiffness and mass matrices, that can be post-processed a-priori if curvature has to be simulated (see subsection 2.2), can be written in the reduced set of coordinates using the projection matrix G defined by Eq. ( 7):

M Cond = G T MG K Cond = G T KG. (9)
The set of retained modal participation factors, P In , can be, then, statically condensed at each frequency step.

Curvature Simulation

Curved structures deserve also some interest. Here a method, to take into account the curvature effects, is presented. With reference to Fig. 2, the idea is to rotate the local reference for each node belonging to the cell FE. This way, imposing the periodicity conditions, as shown in Eq. ( 3) and (4), the wave propagation is automatically analysed along the imposed curved path. Each translational DoF is rotated depending on its distance from the axis of rotation. A single FE model of the cell can be used to simulate different curvatures, and, apart from large periods, the cell can be modelled as totally flat.

In order to achieve the model of the curved cell, a rotational matrix r is defined and assembled in a block diagonal matrix, Rot. It is intended to be done for each curvature and the effects can be superimposed [START_REF] Chronopoulos | A unified approach for the broadband vibroacoustic response of composite shells[END_REF]; in the present analysis, a single curvature is considered. Hence, the mass and stiffness matrices of the curved waveguide are obtained as:

M curv = Rot T M flat Rot K curv = Rot T K flat Rot, ( 10
)
where Rot is the rotation matrix, while the subscript f lat refers to the FE matrices of the periodic cell being modelled as flat and the subscript curv to the ones calculated simulating the curvature of the system. The waves analysed along the locally curved reference (X' in Figs. 2 and3) are circumferential waves. Forcing wavenumbers, imposed after Eq. ( 10), represent, in general, helical waves exciting the semi-infinite cylindrical panel/shell (Fig. 3). It is worth to emphasize that the curvature simulation, showed here, aims at connecting the edge sections of the unit cell through a curved (discrete) system of coordinates. If point of inflections are present, at cell scale, depending on the structure to be described, two approaches are possible, once the cell is modelled with the inflected parts. In one case, the wave propagation can be analysed along the global X-Y axes, simulating a flat waveguide with inflected sections. Otherwise, the cell curvature can be simulated using the nodal coordinates, even the ones belonging to the inflected part of the cell, to evaluate the local rotation of the coordinate system to be simulated; this is translated in a rotation matrix in Eq. ( 10).

Fluid-Structure Coupling

Let us assume a forcing wave impinging on one face of the structure, with an amplitude p I .

The structure, as a result, transmits and reflects waves in the fluid adjacent to the top and bottom surfaces. On the excited side (subscript 1), the sound field is the superposition of the incident and reflected acoustic sound waves, while, in the receiver side (subscript 2), it is given by the transmitted waves.

Assuming the X-Y as the plane of reference (Fig. 1 as example), a forcing acoustic pressure wave can be defined, on the surface of the cell, omitting the time harmonic dependence for simplicity, as (see Fig. 3a):

Flat Surface :

P W = p I e -i(k X X+k Y Y -k Z,1 Z) ; Curved Surface : HW = p I e -i(k X R sin Φ+k Y Y -k Z,1 R cos Φ) ; (11)
where, k X , k Y and k Z are the projections in the global X-Y-Z reference of the plane wave (see Fig. 3a). Here, in the case of curved surfaces, as described also in subsection 2.2, the assumed plane wave is approximated through its projection components on the locally curved surface (Fig. 3b):

HW = p I e -i(k X X +k Y Y -k Z ,1 Z ) ≈ p I e -i(k θ θ+k Y Y -k R Z ) ; (12)
where k X k Y and k Z are the wavenumber components in the new locally rotated reference (see Fig. 3b) and are directly proportional to the circumferential, axial and radial wavenumber components, respectively. It is worth to emphasize that a typical decomposition in cylindrical waves, implying the use of Bessel functions, is not necessary when using a locally rotated reference system as the one in Fig. 3b: Eq. ( 12) depicts helical waves in cylindrical coordinates and the equivalent plane waves in cartesian coordinates, respectively [START_REF] Manconi | Modelling wave propagation in two dimensional structures using finite element analysis[END_REF][START_REF] Manconi | Wave characterization of cylindrical and curved panels using a finite element method[END_REF][START_REF] Manconi | The loss-factor of pre-stressed laminated curved panels and cylinders using a wave and finite element method[END_REF][START_REF] Manconi | Estimation of the loss factor of viscoelastic laminated panels from finite element analysis[END_REF].

From now on, the approximated representation of Eq. ( 12) is used independently on the curvature of the structural model; for infinite radius of curvature (flat structure), the local and global reference system coincide.

If in-plane homogeneous layers are assumed, the local wavenumber components k X and k Y are conserved along the structure, and the only parameter which can vary with the nature of the fluid (or the excitation) is the k Z component, derivable using the Helmholtz equation. When non-homogenised periodic cells are considered, multiple harmonics are added to the k X and k Y terms [START_REF] Mace | Sound radiation from fluid loaded orthogonally stiffened plates[END_REF]. In this framework, the multiple harmonics that arise for periodic non-homogenised structures, are numerically accounted in the structural response of the radiating side (subscript 2).

In fact, when applying the WFE, discrete periodic conditions are applied for the forcing (see Eq. ( 11)) wavenumber couples k X and k Y (see Eqs. ( 2), ( 3) and ( 4)) and for each node subset; the resulting free or forced structural vibration includes any periodicity effects in the frequency band of analysis.

To express the nodal forces on the periodic cell as a function of the pressure amplitudes in the forcing and radiating side of the structure, the dynamic stiffness of the fluids must be derived.

From continuity of the normal particle velocity on the excited and radiating surfaces:

ρ 1 ω 2 q in = ∂(p I -p R ) ∂z ; D f,1 = -iρ 1 ω 2 k Z,1 ρ 2 ω 2 q rad = ∂(p T ) ∂z ; D f,2 = -iρ 2 ω 2 k Z,2 (13)
where ρ 1 and ρ 2 are the fluid densities, q in and q rad are the out-of-plane displacements, respectively of the incident and radiating surfaces, and D f,1 and D f,2 the dynamic stiffness of the fluid in the incident and radiating domains; p I , p R and p T are the incident, reflected and transmitted amplitudes of the sound pressure waves. It is important to notice that, regardless of the homogeneity of the structural model, the derivations over Z, in Eq. ( 13), makes the modelling of the radiating acoustic field somewhat non-sensitive to the presence or absence of multiple harmonics which might arise due to heterogeneity of the structure. These effects are included in the dynamic stiffness of the cell in Eq. ( 5) and are accounted in the structural response.

The load imposed on the plate, by the forcing surface waves of trace wavenumbers k X and k Y , can be derived from the two pressure fields, on both sides of the structure; it is lumped on the wetted nodes of the finite element model. As the forces act normal to the surfaces, the only excited degrees of freedom are the ones connected to the out-of-plane displacements. These ones are identified with the subscript T (top) and B (bottom), while all other degrees of freedom (not excited) are identified by I (internal). The vector of the external forces can be written as:

     e red T e red I e red B      =      S • (p I + p R ) 0 S • p T      ( 14
)
where S is vector of the free nodal surface of each excited node and p I , p R and p T are the nodal pressure vectors. A finer way to calculate consistent nodal forces, requires, however, the knowledge of the shape functions associated with the out-of-plane displacements [START_REF] Yang | Prediction of sound transmission through, and radiation from, panels using a wave and finite element method[END_REF]. The dynamic stiffness matrix and the reduced displacement vector can be rearranged in the same way as in Eq. ( 14), then an energetic equivalence through-thickness applied, condensing all the non-excited nodes [START_REF] Parrinello | Transfer matrix representation for periodic planar media[END_REF].

Including the relation of Eq. ( 13), the dynamic problem results in: where the superscript c indicates that the original D S matrix (Eq. ( 5)) is condensed for the non excited (I) degrees of freedom, through the thickness. The algebraic system in Eq. ( 15) can be solved in p R and p T obtaining the power transmission coefficient τ associated with the couple of forcing wavenumbers k X and k Y .

  D c S T T D c S T B D c S BT D c S BB     p I -p R p T   =   S • D f,1 • (p I + p R ) S • D f,2 • (p T )   (15)
τ (k X , k Y ) = (k Z,2 /ρ 2 )S|p 2 T | (k Z,1 /ρ 1 )S|p 2 I | . ( 16
)
Finite size effects, can be included through correction factors, in order to increase the accuracy at low frequencies. While a formal and accurate spatial windowing approach is present in literature [START_REF] Rhazi | A simple method to account for finite size effects in the transfer matrix method[END_REF], the computational cost associated with this step might be high. For this reason, even losing some accuracy in the low frequency bandwidth range, the use of asymptotic formulas, as in [START_REF] Leppington | The acoustic radiation efficiency from rectangular plates[END_REF],

is here used to reduce the computational cost.

Stochastic Load Translation into Surface Waves

The sound transmission to plane wave excitation, as discussed in Sec. 2, is not sufficient for many applications. Herein, a method, to take into account a general type of excitation, is proposed.

An illustration is reported in Fig. 4. The idea is to use a forcing surface wave excitation for each couple of forcing wavenumbers k X and k Y , able to represent the desired excitation, once its wall pressure spectra, in the wavenumber domain, is known. Using a vectorial form and omitting the harmonic dependence for the sake of readability, a sum of wall plane waves can be written as:

P ( X) = N W j=1 A j e -i Kj X . ( 17
)
where X stands for the couple of surface coordinates (i.e. X-Y in Fig. 1), Kj is the wavenumber vector associated with each wall surface wave of amplitude A j , and N W is the total number of waves constituting the pressure field. It is worth recalling that the surface coordinates can be the global coordinates, if the structure is flat, or, equivalently, the local coordinates if a curvature is present. In the first case, Eq. ( 17) is physically representative of the classic sum of plane waves.

In the other case, it reports a sum of wall helical waves on the curved cylindrical structures.

The description of the desired load is based on the knowledge of the proper values of Kj and A j (Eq. ( 17)). They are obtained by equating the wavenumber spectra φ P P , of the pressure field (Eq. ( 17)), with the one of the fluid excitation model to be simulated, Φ P P , for a specific fluid wavenumber K. Many fluid excitation models are investigated in the wavenumber domain in literature [START_REF] Graham | A comparison of models for the wavenumber-frequency spectrum of turbolent boundary layer pressures[END_REF]. The cross correlation of the pressure field P is:

R P P ( ∆ X 1 , ∆ X 2 ) = N W j=1 A 2 j e -i Kj ( ∆ X1-∆ X2) + N W j=1 N W n=1;n =j A j A n e -i( Kj ∆ X1-Kn ∆ X2) (18)
where the auto and cross correlations have been divided in two different summations. Performing the Fourier transform of Eq. ( 18), the wavenumber spectra is obtained:

φ P P ( K, ω) = Φ P P ( K, ω) = 1 4π 2 N W j=1 A 2 j e i( Kj -K)( ∆ X1-∆ X2) i( Kj -K) +∞ -∞ + N W j=1 N W n=1;n =j A j A n e i( Kj -K)( ∆ X1) -i( Kj -K) +∞ -∞ e i( Kn-K)( ∆ X2) -i( Kn -K) +∞ -∞ . ( 19
)
The expressions in Eq. ( 19) are known and can be written as Dirac delta-functions. Moving from summations and after some algebra, the final expression of φ P P is given in Eq. ( 20). The second term in Eq. ( 20) can be erased since, by definition, the correlation indices j and n must be different while the integration of the product of the Dirac functions is not null only for Kj = Kn .

φ P P ( K) = Φ P P ( K) = 4π 2 A 2 j ∆ Kj δ( Kj -K)d Kj + 8π 4 A j A n ∆ Kj ∆ Kn δ( Kj -K)δ( Kn -K)d Kj d Kn (20)
The amplitudes and the wavenumbers of the simulated waves, able to describe a fluid excitation with the wavenumber spectra Φ P P , are straightforwardly obtainable from Eq. [START_REF] Manconi | The loss-factor of pre-stressed laminated curved panels and cylinders using a wave and finite element method[END_REF]. For each forcing k X and k Y , the desired loading model is simulated through the following surface wave:

P W (X, Y, ω) = Φ P P (k X , k Y , ω)∆k X ∆k Y 4π 2 e -i(k X X+k Y Y ) . (21)
In conclusion, at each frequency and for each wavenumber of the fluid excitation model, a surface wave of specific wavenumbers and amplitudes can be used to simulate the load. No hypothesis on the correlation of the forcing waves is imposed as for the reference coordinates. The proposed approximation is thus valid for correlated and uncorrelated loads acting on plane and curved surfaces, independently on their complexity.

Sound Transmission Loss

In order to simulate the sound transmission for a specific excitation, an integration of the transmission coefficient is performed, in the wavenumber domain. Thus, the total transmission coefficient can be calculated as follows: where W A is the element corresponding to the wall surface wave of wavenumbers k X and k Y , in the matrix of the normalized amplitude functions of all the wavenumber couples involved in the integration process. Within the framework of sound transmission, as identifiable from Eq. ( 16), the variation of waves amplitudes does not have an influence. The difference among different fluid loads is given by the weighting functions W A being involved in the integration. The convergence of the method is assured by the convergence of the integration process. The choice of the integration limits can be changed depending on the type of load to be described. For example, for a diffuse acoustic field, at each frequency step, the wavenumber spectra Φ P P is null outside the acoustic border, so, there is no need for the use of higher integration limits. The final transmission loss is, by definition:

τ T OT (ω) = τ (k X , k Y ) × W A (k X , k Y , ω)dk X dk Y W A (k X , k Y , ω)dk X dk Y (22)
TL(ω) = -10 log 10 (τ TOT (ω)) (23)
The TL expression in Eq. ( 23) includes the assumption, if a curvature is simulated, that the excited surface is a plan projection of the shell surface. The transmission coefficient in Eq. ( 22), must be multiplied by the ratio of cylindrical section and its plan projection. However, in the case of curved finite structures, instead of using a baffled window equivalence, for accounting the effects of the finiteness of the structure, an alternative approach based on the ratio of the excited surfaces is here proposed. With reference to Fig. 5, the ratio between the area of the equivalent cylindrical portion (A cyl ), built starting from the finite curved panel analysed, and the effectively excited area (A exc ), is multiplied, in a SEA (Statistical Energy Analysis) fashion, to the transmission coefficient in Eq. [START_REF] Christen | Wave finite element formulation of the acoustic transmission through complex infinite plates[END_REF]. The approach is consistent since, given a certain length of the panel, along the non-curved side, its area can increase just up to the one of the equivalent cylindrical portion. The resulting TL, thus, can asymptotically converge toward the one of the equivalent cylinder. In these cases, Eq. ( 23) becomes:

TL(ω) = -10 log 10 τ TOT (ω) πA cyl 2A exc . ( 24
)
The advantage of the present approach relies in its generality and applicability to a wide range of test-cases both in terms of structural shapes and excitation models. The wall surface wave approximation releases the constraints to the plane waves angles of incidence, generally implied in literature, widening the analysable combinations of structural and excitation models. On the structural point of view, the only requirement, for the applicability of the present approach, is given by the homogeneity or periodicity, independently on the reference direction (flat or curved).

On the other hand, for the simulated excitation, the basic requirement is the knowledge of the wall pressure spectrum.

Validation for Diffuse Field Transmission

In this section, a series of comparisons are presented for validation purposes. Both flat and curved composite sandwich panels are analysed. A validation for cylinders is also present, under diffuse acoustic loading. In all the test-cases proposed here, the dispersion curves in the in the X and Y direction (Fig. 1,2) will be provided in order to observe acoustic coincidences in terms of waves.

Flat Homogenised Panel

In Fig. 6, a comparison among the proposed method and two other approaches available in literature is shown [START_REF] Christen | Wave finite element formulation of the acoustic transmission through complex infinite plates[END_REF][START_REF] Yang | Prediction of sound transmission through, and radiation from, panels using a wave and finite element method[END_REF]. The analysed sample is a 3mm thick flat sandwich panel made of 1mmthick aluminium skins and a 1.5mm-thick isotropic core (E = 3 GPa, ν = 0.2, ρ = 48 Kg/m 3 ).

The cell is modelled using four ANSYS solid elements through thickness. The dispersion curves in Fig. 6a, show an acoustic coincidence at ≈ 6 kHz. In Fig. 6b an excellent agreement is observed for the sound transmission loss calculated using the present approach and the numerical ones in [START_REF] Christen | Wave finite element formulation of the acoustic transmission through complex infinite plates[END_REF][START_REF] Yang | Prediction of sound transmission through, and radiation from, panels using a wave and finite element method[END_REF]. Finite size effects are not included in the model and thus the comparisons in Fig. 6 are for infinite panels.

Flat Heterogeneous Panels

The second test-cases consist in an aluminium double-wall flat panel with mechanical connections (Fig. 7a), and a sandwich panel with rectangular core (Fig. 7b) made in ABS (E = 1.8 GPa, ν = 0.35, ρ = 998 Kg/m 3 ). The dispersion curves of the two test-cases analysed are plotted in Fig. 8 with the acoustic wavenumbers versus frequency. The wavenumbers are derived from the propagations constants which are solutions of the eigenvalue problem in Eq. ( 5). Respectively, Fig. 9a shows the comparison for the TL, using an in-house reproduced code of the method presented by Christen et al. [START_REF] Christen | Wave finite element formulation of the acoustic transmission through complex infinite plates[END_REF],

whereas, Fig. 9b, shows the comparison for the TL of the sandwich rectangular-cored panel using, as a reference, the transfer matrix approach proposed by Parrinello et al. [START_REF] Parrinello | Transfer matrix representation for periodic planar media[END_REF]. The double-wall cell is 5mm thick, 10mm long in the periodic direction (X), 1mm long in the homogeneous direction (Y)

and the walls have a thickness of 1mm. The rectangular cored sandwich cell is 10mm thick, 10mm long both in X and Y, the skins and the core walls have a thickness of 0.6mm. Finite size effects are not included in the model and thus the comparisons in Fig. 6 are for infinite panels. Again excellent agreement is observed, even for this complex structural shape, validating the proposed approach also for large heterogeneity scales and non-homogenised structure models. In this case, ANSYS shell elements are used instead, for the FE modelling. In all presented results, the mesh used for the calculations is verified to converge in the frequency band investigated.

Curved Homogenised Panels and Cylinders

The presence of curvature induces an alteration in the structural behaviour, at least, up to the ring frequency. This is the eigenfrequency corresponding to the first extensional mode at which the longitudinal wavelength is equal to the circumference of the structural element. At this frequency, the shell sound radiation is amplified similarly to a coincidence condition [START_REF] Koval | On sound transmission into an orthotropic shell[END_REF]. The transmission of curved panels and cylinders is here validated, under diffuse acoustic load, using, as a reference, numerical and experimental data available in literature.

The shell is a 3mm thick aluminium one and the curvature radius is 2m and the dispersion curves versus the acoustic wavenumbers are reported in Fig. 10a. First, in Fig. 10b, the transmission loss of an infinite cylinder is compared to the numerical work by Ghinet [START_REF] Ghinet | Diffuse field transmission into infinite sandwich composite and laminate composite cylinders[END_REF]. Both the ring frequency (≈ 400 Hz; see Fig. 10a) and the acoustic coincidence (≈ 4.0 kHz; see Fig. 10a) are accurately described and the overall agreement is excellent. Loss comparison with numerical results form [START_REF] Ghinet | Diffuse field transmission into infinite sandwich composite and laminate composite cylinders[END_REF]. 1.

Fig. 11b shows the sound transmission loss for a 2x2.4m 2 sandwich composite panel whose skin and core are, respectively, 1.2mm and 12.7mm thick. The present method leads to a very good agreement even in the low frequency range. Again, both the ring frequency (≈ 400 Hz) and the acoustic coincidence (≈ 1.0 kHz), observable from the dispersion curves in Fig. 11a, are well predicted.

The proposed validations demonstrate the accuracy for both the methodology to account for curvature effects and the proposed approach to include finite size effects in the case of curved structures (Eq. ( 24) is used here) .

Curved Complex Panel

In this subsection a final test-case is studied: a doubly-ribbed curved fuselage panel with fames and stringers. The structure is characterised by a strong heterogeneity and complexity and sound transmission loss measurements, from [START_REF] Orrenius | Wave modelling in predictive vibro-acoustics: Applications to rail vehicles and aircraft[END_REF], are used as reference results. The panel has dimensions 1.45 m x 1.70 m, with a 1.35 m radius of curvature; the geometrical parameters are reported in Table 2, while the material is an aluminium alloy for all its substructures [START_REF] Orrenius | Wave modelling in predictive vibro-acoustics: Applications to rail vehicles and aircraft[END_REF]. An illustration of the panel and the relative substructure used within this WFE-based framework, are shown in Fig.

12. It can be observed from Fig. 12, that the real panel has non periodic elements at the borders; some differences between the real structure and the ideally periodic model we assume within this approach, are thus present.

The modelled periodic cell has more than 2.9 • 10 4 degrees of freedom, that are reduced to 6.0 • 10 4 using the modal order reduction described in Section 2. Here, the finite size effects are accounted using the approximation in Eq. ( 24). In Fig. 13, a good agreement is observed in the 300 Hz -2.0 kHz frequency range with the experimental measurements from [START_REF] Orrenius | Wave modelling in predictive vibro-acoustics: Applications to rail vehicles and aircraft[END_REF][START_REF] Orrenius | Analysis of sound transmission through aircraft fuselages excited by turbulent boundary layer or diffuse acoustic pressure fields[END_REF]. Both the numerical and experimental TL curves drop around the ring frequency (≈ 630 Hz; [START_REF] Orrenius | Wave modelling in predictive vibro-acoustics: Applications to rail vehicles and aircraft[END_REF][START_REF] Orrenius | Analysis of sound transmission through aircraft fuselages excited by turbulent boundary layer or diffuse acoustic pressure fields[END_REF]). This inversely proves that, even for such a complex-shaped structure, the curvature simulation presented in Eq. ( 10) is still applicable (L x /R still small enough).

Validation for Boundary Layer Transmission

The load approximation proposed in Section 3, leading to Eq. ( 22), allows the simulation for loads with different characteristics. In this section, for example, the flow-induced transmission caused by turbulent boundary layer (TBL) excitation is considered. This is known for being one of the main sources of radiated noise inside an aircraft cabin, in cruise flight conditions.

While many boundary layer models are proposed in literature, here, the characterisation of the wall pressure fluctuations proposed by Corcos is used [START_REF] Corcos | Resolution of pressure in turbulence[END_REF][START_REF] Graham | A comparison of models for the wavenumber-frequency spectrum of turbolent boundary layer pressures[END_REF]. It is assumed that no-gradients effects are present and the TBL is fully developed. The wavenumber spectra, Φ P P , proposed by Corcos, is here reported for the sake of completeness assuming the directions X and Y as the stream-wise and cross-wise ones:

Φ P P (k X , k Y , ω) = S pp (ω) 4α X α Y α 2 Y + U 2 c k 2 Y ω 2 α 2 X + 1 + Uck X ω 2 (25)
where U c is the convective flow speed, S pp is the single-point auto spectral density of the wall pressure distribution. The stream-wise and cross-wise correlation coefficients, α X and α Y , are assumed to be 0.125 and 0.78, respectively, in all following test-cases.

First, a validation for the boundary layer transmission, in the case of a flat isotropic panel with simply-supported boundary conditions (edges), is proposed in Fig. 14. The reference solution is calculated using a full FE method, as proposed, and validated, in many works in literature [START_REF] Franco | Numerical approximations on the predictive responses of plates under stochastic and convective loads[END_REF][START_REF] Rosa | Exact and numerical responses of a plate under a turbulent boundary layer excitation[END_REF].

The eigen-frequencies and the modal shapes of the reference panel are calculated using analytic solutions, while the load matrix is described using a direct method [START_REF] Franco | Numerical approximations on the predictive responses of plates under stochastic and convective loads[END_REF]. The incident power, for the transmission loss calculation, in the FEM cases, is calculated using Eq. ( 26) as proposed in literature [START_REF] Atalla | Modelling the sound transmission through complex structures with attached noise control materials[END_REF][START_REF] Barisciano | Broadband transmission loss due to reverberant excitation[END_REF]:

Π inc = AS pp (ω) 4ρ 1 c 1 ( 26
)
where A is the excited area and c 1 the speed of sound on the incident side. A modal behaviour can not be described using the present method, as in an SEA framework, since the wave propagation is considered to the infinite (reflections at the borders are neglected) and a semi-infinite fluid termination is assumed (no internal cavity modes). The averaged sound transmission losses are calculated in discrete frequencies (in logarithmic space), while the FEM calculation is in third octaves bands. In both the test-cases proposed here, the dispersion curves in the flow direction (X) is provided in order to show the convective/aerodynamic and acoustic coincidences.

For the aluminium case (Fig. 14), the aerodynamic and acoustic coincidences are at ≈ 1.6 kHz and ≈ 6.0 kHz, as shown in the dispersion curves in Fig. 14a. Here, the finite size effects are accounted using a radiation efficiency formulation for flat panels, as proposed by Leppington [START_REF] Leppington | The acoustic radiation efficiency from rectangular plates[END_REF].

As shown in Fig. 14b, both the aerodynamic and acoustic coincidence dips are correctly identified Differently, an honeycomb-cored sandwich panel is analysed under a turbulent boundary layer in Fig. 15. A Corcos model is used for the loading description and the proposed approximation into surface waves is used. The panel is made of 1mm-thick aluminium skins and a 10mm-thick hexagonal Nomex honeycomb core (material properties in Table 1), homogenised in an equivalent orthotrophic model. Both the aerodynamic and acoustic coincidences are well predicted in the sound transmission loss curves, in Fig. 15b. The first one is somewhat highly damped (≈ 600 Hz; see Fig. 15a) while the second one is clearly visible (≈ 9 KHz; Fig. 15a).

The strong agreement observed in Fig. 14 and 15, validates the proposed load approximation even for spatially-correlated random loads, as the TBL. It is worth underlying how the use of boundary layer excitation is here allowed even for infinite structures, differently from other methods in literature [START_REF] Atalla | Modelling the sound transmission through complex structures with attached noise control materials[END_REF].

The Effective Transmission Loss of Cylinders

In the case of shells, the load is translated in helical waves, instead of wall plane waves. For example, the effective transmission loss of a shell, in the case of simultaneous acoustic and aerodynamic excitation, can be estimated, assuming the axial direction of the shell as the stream-wise and the circumferential as an approximated cross-wise. In Fig. 16 the dispersion curves in the circumferential direction (wavenumbers of purely circumferential waves) are plotted for three different 3.2mm-thick aluminium shells; 3.0, 2.0 and 0.75 m curvature radii are considered and the ring frequencies are, respectively, 280, 400 and 1090 Hz. Differently, in Fig. 17, the effective transmission loss is calculated and compared. The aim, in this case, is to investigate how the aerodynamic coincidence influences the shell transmission.

In the case of the two bigger cylinders (see Fig. 16), the ratio between the ring frequency and the aerodynamic coincidence frequency is lower than one (on the flat case the radius equal to infinite). Right after the ring frequency, when the curvature effects start to vanish, these shells still behave in a sub-convective domain, thus, the effects of the coincidence are clearly visible in the sound transmission loss (see Fig. 17). On the other hand, in the case of the small cylinder, the ring frequency is superior to the critical aerodynamic frequency and the shell behaves in a subconvective domain only when the curvature effects are important. The aerodynamic coincidence peak, in the transmission loss, is no more identifiable.

Similarly, if a single shell is analysed under different convective speeds, the aerodynamic coincidence region moves in accordance to what happens in the operative conditions of a transport system as an aircraft, a train or a vehicle. The boundary layer effect vanishes when the speed lowers, since, the convective critical frequency lowers, up to getting inferior to the ring frequency.

Discussion on approximations and limits

It is useful to discuss and emphasize some aspects regarding the approximations and the limits of the present numerical method, presented in Sections 2 and 3, using as reference and illustrative schemes Figs. 1, 2, 3, 4 and 5.

The curvature simulation in Eq. ( 10), is a good approximation for homogenised models. For periodic non-homogenised models, relatively small curvatures can be simulated. In fact, when the ratio between the length of the cell (in the curvature direction) and the radius of curvature is not small enough, the piecewise-flat approximation used here is critical even for fine structural meshes.

In addition, great care must be placed on not curving even potential internal elements which can be flat (i.e resonator beams).

The interior acoustic field, as illustrated in Fig. 4, is assumed to be equivalent to a semiinfinite fluid termination: the modal behaviour of a finite cavity can not be described in this context, even for a closed cylindrical model. In analogy to what happens for an infinite flat structure, the internal acoustics is assumed to be composed by single out-going waves and internal acoustic waves' reflections/transmission are not modelled (see Eq. ( 13)). These are the same assumptions/approximations used in [START_REF] Ghinet | The transmission loss of curved laminates and sandwich composite panels[END_REF][START_REF] Ghinet | Diffuse field transmission into infinite sandwich composite and laminate composite cylinders[END_REF].

The method provides good results, as expected, in the limit where the periodic length is small compared to structural and acoustic wavelengths: homogenised (or smeared) models are enough for a correct description of the wave-guide. Nevertheless, the present comparisons, especially in Fig. 13, show a more general applicability well beyond the homogenisation limits and for large periodic scales. In fact, in this framework, the multiple harmonics that arise in the radiated acoustic field, for purely periodic (non-homogenised) structures, are numerically included in the structural response of the radiating side, when applying discrete periodic conditions for each couple of forcing wavenumbers k X and k Y . It, coherently to what has been discussed for the semi-infinite fluid termination condition, does not need to be analytically exploited since the only useful factor, in this method, for the acoustics in the radiating side is the k Z term, dependent on the nature of the fluid itself (see Eq. ( 13)). In fact, when comparing the present approach with the work by Ghinet et al. [START_REF] Ghinet | Diffuse field transmission into infinite sandwich composite and laminate composite cylinders[END_REF], for an infinite isotropic cylinder under a diffuse acoustic field the agreement is excellent in the whole frequency band; the reference method, in [START_REF] Ghinet | Diffuse field transmission into infinite sandwich composite and laminate composite cylinders[END_REF], which is semi-analytical, properly describes the acoustic field in global coordinates, with a full development using spherical harmonics.

Moreover, the structural wave propagating in the periodic media is here considered to the infinite. In other words, waves' reflections, transmission and absorption, typical at the borders of a finite media, are not accounted and, thus, a single-modal behaviour can not be described with this model. However, the finite size effects, which help in re-scaling the sound transmission loss versus frequency (with respect to the one of an infinite structure) can be included in the model using correction factors, as described in [START_REF] Villot | Predicting the acoustical radiation of finite size multilayered structures by applying spatial windowing on infinite structures[END_REF].

Regarding the load approximation, the use of a lumped-on-the-nodes method, causes aliasing when the nodal areas are not small-enough to assume negligible pressure fluctuations within [START_REF] Franco | Numerical approximations on the predictive responses of plates under stochastic and convective loads[END_REF][START_REF] Rosa | A scaling procedure for the response of an isolated system with high modal overlap factor[END_REF].

While this approach is used because of its flexibility (the shape functions of the models are not required), it requires some care for convergence aspects. There are two main steps to check for a proper convergence. First, at cell scale, the single elements should have proper dimension to be able to describe the smallest wavelengths at the maximum frequency of analysis. This is a well-known mesh sizing problem. In the present work, more than 6 (in some cases more than 10) elements per minimum wavelength have been used. Here, the approximation described in Section Once the structural mesh sizing is performed, the integration scheme in Eq. ( 22) must be properly carried out. Depending on the wavenumber spectra of the excitation function, a proper mesh in the wavenumber space has to be used to describe the function Φ P P (k X , k Y ), at each frequency step. This is a typical discretization problem for functions of two variables. However, while an indiscriminate increase of the wavenumber sampling results in a more accurate integration, a strong increase of the number of operations can lead to high computational cost. A trade-off solution resulting from a convergence study, is strongly suggested. In Fig. 18, an example concerning the wavenumber integration is shown. In Fig. 18a a coarser mesh (wavenumber sampling with a 50x50 mesh) is used and, as a result, the TL predictions are not as accurate as expected. In Fig. 18b, on the contrary, a finer integration (wavenumber sampling with a 150x150 mesh) is applied and the results are very accurate broadband. This time, the reference solution is the same proposed in Fig. 14, in twelve octave bands.

A general rule is hard to define since the wavenumber spectra might differ widely, depending on the nature of the excitation. General numerical rules for the discretization of two-dimensional functions must be used case-by-case and a pre-calculation convergence study is suggested.

Conclusions

This work proposes a numerical approach for the estimation of the sound transmission loss of complex flat, curved and cylindrical periodic structures, under any type of acoustic and aerodynamic load. The approach involves a wave finite element method, for the structural part, and proposes a load simulation into surface waves. The fluid-structure interaction is performed in analogy to the acoustic wave excitation, discriminating among the different forcing models, using a weighted wavenumber integration. The only requirement is the knowledge of the wavenumber spectra of the wall pressure fluctuation. Finite size effects are accounted using the baffled window equivalence or asymptotic formulations, for flat structures. An alternative and efficient method is proposed in the case of curved finite structures, in similitude to the semi-infinite equivalent cases.

Static and dynamic condensation can be applied if fine meshes are used in the modelling phase.

Both the accuracy and robustness of the present method are proved using analytic, numerical and experimental references. Both uncorrelated (diffuse acoustic field) and spatially-correlated loads (turbulent boundary layer) are used for the validations, in the case of flat and curved structures. Calculations performed using finite elements of different nature do not affect the accuracy of the estimations. The convergence of the approach is assured by the one of the wavenumber integration process. The choice of the integration limits must be calculated on the base of the wavenumber spectra of the load.

The combination of boundary layer and acoustic excitation is simulated in the case of shells, resulting in an effective sound transmission loss. The ring frequency, the aerodynamic and acoustic coincidences are efficiently estimated, independently on the curvature radius and convective velocity simulated. Moreover, the use of boundary layer excitation does not require the accounting of finite size effects and a comparison of the structural and acoustic design is possible, independently from the size of the analysed structure.

Figure 1 :

 1 Figure 1: Example of a FE cell model with periodicity along the X-Y directions. a) Isometric view; b) Top view with nodes' subsets.

Figure 2 :

 2 Figure 2: Rotation of the local system of reference for each node of the periodic cell FE model

Figure 3 :

 3 Figure 3: Global and Local reference systems used for waves along curved structures and shells. a) Global Cartesian reference; b) Local Curved Reference.

Figure 4 :

 4 Figure 4: Illustration of the load simulation using surface waves in a general test-case. a) Real/Physical situation; b) Simulated case.

Figure 5 :

 5 Figure 5: Excited surface comparison between a curved finite panel and its equivalent cylindrical portion.

Figure 6 :

 6 Figure 6: A sandwich flat panel under diffuse acoustic field excitation. a) Dispersion curves; b) Transmission Loss comparison with literature references ([22, 23]) in a logarithmic frequency step, between 100 Hz and 10KHz.

Figure 7 :

 7 Figure 7: Portion of a double wall flat panel with structural links and the detail of the unit cell analysed

Figure 8 :

 8 Figure 8: Dispersion Curves for the sandwich panel designs in Fig. 7. The wavenumbers represent derive from the eigenvalues of Eq. (5): a) Double-wall panel with structural link; b) Sandiwch panel with rectangular core.

Figure 9 :

 9 Figure 9: Transmission Loss comparison for the sandwich panel designs in Fig.7, under diffuse acoustic field excitation. A logarithmic frequency step, between 100 Hz and 10KHz, is used to generate the curve labelled as P resent M ethod. a) Reference method in[START_REF] Christen | Wave finite element formulation of the acoustic transmission through complex infinite plates[END_REF]; b) Reference method in[START_REF] Parrinello | Transfer matrix representation for periodic planar media[END_REF]

Figure 10 :

 10 Figure 10: An isotropic cylinder under diffuse acoustic field excitation. a) Dispersion curves; b) Sound Transmission

Figure 11 :

 11 Figure 11: A finite curved sandwich composite panel under diffuse acoustic field excitation. a) Dispersion Curves comparison with results from [29]. b) TL comparison with measurements from [29] are compared with the actual method, in third octave bands.

Figure 12 :

 12 Figure12: The panel tested in[START_REF] Orrenius | Analysis of sound transmission through aircraft fuselages excited by turbulent boundary layer or diffuse acoustic pressure fields[END_REF] and the unit cell used for the WFE simulation: cell sizes are coherent with the stiffeners spacing in Table2.

  .

Figure 13 :

 13 Figure 13: The Sound Transmission Loss of a curved ribbed panel under diffuse acoustic field; comparison with numerical and experimental results form [24].

Figure 14 :

 14 Figure 14: An aluminium plate under TBL excitation: a) Dispersion curves with convective and acoustic wavenumbers; b) TL numerical comparison with full FEM -Uc = 180 m/s; A = 0.5x0.3m 2 . A logarithmic frequency step, between 100 Hz and 10KHz, is used to generate the curve labelled as P resent M ethod.

Figure 15 :

 15 Figure 15: An honeycomb sandwich plate under TBL excitation: a) Dispersion curves with convective and acoustic wavenumbers; b) TL numerical comparison with FEM method -Uc = 152 m/s; A = 0.8x0.6m 2 . A logarithmic frequency step, between 100 Hz and 10KHz, is used to generate the curve labelled as P resent M ethod.

Figure 16 :

 16 Figure 16: Dispersion curves in the circumferential direction of shells of different curvature. The acoustic and convective (Uc = 185 m/s) wavenumbers are shown.

Figure 17 :

 17 Figure 17: The effective transmission loss for shells of different curvature -Uc = 185 m/s. A logarithmic frequency step, between 100 Hz and 5KHz, is used to generate the curves.

Figure 18 :

 18 Figure 18: The transmission loss of an aluminium panel under TBL excitation; numerical comparison with full FEM method in octaves -Uc = 180 m/s; A = 0.7x0.5m 2 . a) Coarse meshing of the TBL wavenumber spectra; b) Fine meshing of the TBL wavenumber spectra

Table 1 :

 1 Materials' properties for curved finite panels transmission loss validation

		Skin	Core
	E 1 (GPa)	48.0	0.145
	G 1,2 (GPa)	18.1	0.05
	G 1,3 (GPa)	2.75	0.05
	ν 1,2	0.3	0.2

ρ (Kg/m 3 ) 1550.0 110.44 On the other hand, in Figs. 11, a validation versus experimental measurements for curved finite panels is shown. In both cases the curvature radius is 2m; the skin and core material are reported in Table

Table 2 :

 2 Geometrical parameters of the ribbed fuselage panel.

		Frames Stringers Skin
	Thickness (mm)	1.8	1.2	1.2
	Height (mm)	72	28
	Spacing (mm)	40.6	15.2

Acknowledgments

This project has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 675441. The author would like to gratefully acknowledge everyone involved in the VIPER project.