Probabilistic approach of uncertainties for the computational model and identification

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Computational model for the vibroacoustic problem

A computational model is constructed to solve Eqs. (2.1) to (2.50) using the finite element method. Primary variables for the description of the structure and of the internal acoustic fluid repectively are the displacement field us and the acoustic fluid pressure disturbance field p. Primary variables for the description of the poroelastic medium are the solid and fluid phases displacements fields ups and upf . Ensuing, we have compatible finite element meshes of s, f and p and those fields are interpolated on their nodal values by a finite element basis.

Finite element implementation

The whole finite element procedure as well as the mesher are implemented from scratch in Matlab such that a limited array of input parameters controls the construction of the computational model. Three dimensional eight nodes volumic elements as well as two dimensional four nodes shell elements with normal rotational degree of freedom are used for the discretization of the sesquilinear form associated with the structure. Those elements are constructed following the references [72–74] in which specific reduced integration of the shear stress with respect to the hexahedron element and a modified flexibility with respect to the shell elements allow to avoid the di erent locking phenomena inherent to elements with low degree interpolating polynomials. The finite elements involved in the discretization of the acoustic fluid and poroelastic fluid phase are the classical eight nodes isoparametric fully integrated ele-ments. In particular, the finite element implementation with respect to the Biot displacement formulation is validated in Appendix A according to the cases presented in [35] and previously introduced in [75].

Definition of the admissible set for the structural system parameters

The structure, constituted of the assembly of di erent elements supposed homogeneous, is described through the knowledge of parameters associated with its geometry, elasticity tensors and mass densities. Elements of vector p associated with materials elastic properties are then typically Young’s moduli E j, shear moduli G jk, Poisson’s ratios jk. Let CE , CG and C respectively be the admissible set for the three kind of elastic parameters. Positiveness of the strain energy implies the strict positiveness of Young’s moduli as well as of shear moduli such that CE =]0; +1[ and CG =]0; +1[. For isotropic materials, Poisson’s ratio are bounded and we have C = [ 1; 0:5]. Within the general anisotropic case, Poisson’s ratio is not bounded [82] and belong to R, with the constraint that the strain energy remains positive. Hereinafter, given the considered class of materials, the support of Poisson’s ratios is chosen such that C = R. Moreover, the mass density s belong to C =]0; +1[. Consequently, within the case of elastic parameters identification, the admissible set Cpar for the vector of the system parameters p is defined by Cpar = n p 2 Rnp j 8 j; k; E j 2 CE ; G jk 2 CG; jk 2 C ; s 2 C o : (3.1).

Definition of the optimisation problem

A computational model is constructed for the frequency band B = [!min; !max], according to Sec-tion 2.4, and parametrized by the vector p belonging to Cpar subset of Rnp . For nex excitation points, nobs mobility functions are experimentally measured and subsequently computed. According to the usual notations, y jk(!) denotes the mobility associated with the point k when the point j is submitted to an excitation. The parameter dependent error estimator p 7!Ep(p) is then defined by nex nobs Z !max j log10 jy jk(!; p)j log10 jymesjk(!)j j2 d! ; (3.2) Ep(p) = j=1 k=1 !min.

Experimental identification of mounting parameters

It should be noted that the run through the present approach was carried out in reverse: given an assembled shear panel fresh out the manufacture, measurements were carried out and the system gradually dismantled in order to perform dierent sets of measurements focusing on dierent identification problems.
This also explains why, in Section 4.4, the properties of only one board could have been identified due to the delicate and destructive take o of the boards from the beam structure. Consequently, in the following, we first endeavour to model an assembly of beams and then the assembly of one  board on beams in order to translate the obtained information to the whole assembled shear panel.

Assembly of one oriented strand board and beam elements

In order to investigate such potential flexible eects in regard to board onto beam assemblies, as well as to assess for the latter possibility to be taken into account with the previous model, a simple system is designed, built, and experimental measurements are carried out. A suspended assembly of one oriented strand board on top of four wooden beams (see Fig. 4.3), of dimensions 2:405 m 0:635 m 0:152 m, is excited on dierent points and experimental mobilities are measured over the frequency band B = [10; 280] Hz. Beams 1 and 2 are of dimension 2405 mm 45 mm 140 mm while beams 3 and 4 are of dimension 45 mm 545 mm 140 mm and the board has a thickness of 12 mm. A computational model is constructed for the frequency band B, in which the structural loss factor is once gain fixed to s = 0:02. The dierent physical properties, identified from experimental measurements according to the methodology introduced in Chapter 3, are given in Tables 4.4 and 4.5. Noteworthy is the fact that the oriented strand board has not the same origin than those treated in Section 3.3.2, thus explaining the large discrepancies between identified elastic properties and mass densities for the former, given in Table 3.7 with respect to those of the latter. Unfortunately, only one board specimen could have been measured but, given the range of properties observed in Section 3.3.2, the considered oriented strand board is most likely not a realisation from the same product.

Whole assembled shear panel

In Sections 4.3.1 and 4.3.2, parameters associated with the flexible model were identified for dierent configurations. In this paragraph, the acquired information is applied in the case of a whole assembled shear panel, constituted of oriented strand boards mounted on an assembly of beams. The beam assembly treated in Section 4.3.1 is now covered with five oriented strand boards, delimited by the grey areas on Fig. 4.5. Two new observation points, whose coordinates are given in Table 4.8, are placed on the boards such that accelerometers on each side of the system provide transfer mobilities from boards to boards and boards to beams.
A computational model is constructed for the frequency band B = [10; 280] Hz, in which the structural loss factor is fixed to s = 0:02. The dierent physical properties are given in Tables 4.1 and 4.5. Each oriented strand board is modeled with an identical mass density and elasticity tensor. Boards are however not connected with eachother on the edges, as an air gap of the order of 5 mm is observed between them. Mounting parameters for the beam assembly are those identified in Section 3.3.2 and given in Table 4.2, which are eectively identical as the system is the same, while the mounting parameter associated with the board onto beam connections is the one identified in Section 4.3.2 and given in Table 4.6 which is chosen as a prior nominal value because none of the board onto beam connections of the systems exactly corresponds to the ones treated within the previous section. Figure 4.2 displays the squared modulus of the mobilities associated with the velocities in the z direction on observations points 1, 7, 8 and 9 for an excitation in the z direction on points 7 and 8. First of all, for any of the observation point, the model taking into account the flexible mounting improves the prediction in comparison with the model using perfectly tied connections. However, the input mobility y88, located on a board straight on top of a central beam, shows that the updated model fails to give a good prediction for the points located directly on the connections, starting from 100 Hz approximately.
According to the french standards ( [102] in regard to walls and [103] in regard to floors) the boards were nailed every 15 cm to the sidelong beams and 30 cm to the central beams. Moreover, an approximation of the bending wavelength, function of the frequency, can be computed using the analytical dispersion relation associated with an a isotropic thin plate given in [81] such that k4 = !2 sh B .

Table of contents :

Introduction 
1.1 Context
1.2 Objectives
1.3 Strategy
1.4 Positioning
1.5 Structure of the document
2 Vibroacoustic problem 
2.1 Introduction
2.2 Definition of the vibroacoustic system
2.3 Boundary value problem for the vibroacoustic system
2.3.1 Structure
2.3.2 Internal dissipative acoustic fluid
2.3.3 Poroelastic medium
2.4 Computational model for the vibroacoustic problem
2.4.1 Finite element implementation
2.4.2 Computational model
2.5 Alternative computational model with limp frame poroelastic medium modeled as an equivalent fluid
2.6 Conclusion
3 Overview of lightweight structural materials and identification from experimental measurements
3.1 Introduction
3.2 System parameters identification problem
3.2.1 Definition of the admissible set for the structural system parameters
3.2.2 Definition of the optimisation problem
3.3 Validation of the computational model and identification of the elastic parameters for typical lightweight building elements
3.3.1 Wooden beams
3.3.2 Lightweight boards
3.4 Conclusion
4 Flexible connection model and experimental identification 
4.1 Introduction
4.2 Computational model for flexible mounting
4.3 Experimental identification of mounting parameters
4.3.1 Assembly of beam elements
4.3.2 Assembly of one oriented strand board and beam elements
4.4 Whole assembled shear panel
4.5 Conclusion
5 Reduced order computational model for the vibroacoustic problem 
5.1 Introduction
5.2 Reduced order model for the structure
5.2.1 Construction of the truncated projection basis
5.2.2 Generalized matrices for the reduced order model
5.3 Reduced order model for the internal acoustic fluid
5.3.1 Construction of the truncated projection basis
5.3.2 Generalized matrices for the reduced order model
5.4 Reduced order model for the poroelastic medium modeled as coupled solid and fluid phases with displacements as primary variables
5.4.1 Construction of the truncated projection basis
5.4.2 Generalized matrices for the reduced order model
5.4.3 Comparison of dierent reduction strategies for a poroelastic medium coupled with an acoustic cavity
5.5 Reduced order model for the poroelastic medium modeled as an equivalent fluid with pressure as the primary variable
5.5.1 Construction of the truncated projection basis
5.5.2 Generalized matrices for the reduced order model
5.6 Assembled reduced order computational models
5.7 Conclusion
6 Probabilistic approach of uncertainties for the computational model and identification 
6.1 Introduction
6.2 Stochastic computational model and uncertainty quantification
6.2.1 Stochastic reduced order computational model
6.2.2 Convergence of the random solution
6.2.3 Confidence regions for the observables
6.3 Generalized probabilistic approach of uncertainties
6.4 Probabilistic approach of system parameters uncertainties
6.4.1 Prior probabilistic model of uncertainties for the structure parameters
6.4.2 Strategies for the identification of the prior probabilistic model hyperparameters from experimental measurement 6.4.3 Identification of the prior probabilistic model hyperparameters for typical structural lightweight components
6.5 Uncertainty quantification for a shear panel and comparison with experimental measurements
6.5.1 Mean model taking into account flexible connections
6.5.2 Mean model including modeling errors induced by perfectly rigid connections .
6.6 Conclusion
7 Airborne sound insulation 
7.1 Introduction
7.2 Model for the evaluation of airborne sound insulation
7.2.1 Parallelepiped room model
7.2.2 Analytical modal expansion of the pressure field in the rooms
7.2.3 Decoupled approach for the evaluation of the sound reduction index
7.2.4 Concluding remarks about the approach
7.3 Application to double parting wall separative systems
7.3.1 Nominal systems
7.3.2 Mean computational models
7.3.3 Definition of the external excitation for the computational model
7.3.4 Evaluation of the sound reduction indices
7.3.5 Uncertainty quantification
7.4 Conclusion
8 Impact sound insulation 
8.1 Introduction
8.2 Model for the evaluation of impact noise level
8.3 Model for the tapping machine excitation force
8.3.1 Probabilistic model for the external excitation resulting from the tapping machine 101
8.3.2 Concluding remarks about the approach
8.4 Validation of the computational model for the impact problem and uncertainty quantification
8.4.1 Experimental validation of the computational model for a simple lightweight system
8.4.2 Uncertainty quantification
8.4.3 Concluding remarks about impact forces modeling
8.5 Application to a full scale lightweight floor system
8.5.1 Nominal system
8.5.2 Mean steady-state computational model
8.5.3 Stochastic steady-state computational model
8.5.4 Comparison with experimental measurements: velocity levels
8.5.5 Comparison with experimental measurements: impact sound level
8.6 Conclusion
9 Optimisation 
9.1 Introduction
9.2 Optimisation algorithm
9.2.1 Definition of the fitness functions
9.2.2 Genetic algorithm
9.3 Robust optimisation problems
9.3.1 Lightweight double parting wall systems
9.3.2 Lightweigth floor system
9.4 Conclusion
Conclusions and perspectives
Appendices
A Validation of the finite element implementation with respect to the Biot displacement formulation
A.1 Analytical solutions for the sound propagation in a unidimensional poroelastic
medium
A.2 Unidimensional reference problems
B Limp frame poroelastic medium equivalent fluid model
References 

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