Dynamic substructuring of damped structures using singular value decomposition Roger Ohayon, R. Sampaio, Christian Soize
To cite this version:
Roger Ohayon, R. Sampaio, Christian Soize. Dynamic substructuring of damped structures using sin- gular value decomposition. Journal of Applied Mechanics, American Society of Mechanical Engineers, 1997, 64 (2), pp.292-298. 10.1115/1.2787306. hal-00770023
HAL Id: hal-00770023 https://hal-upec-upem.archives-ouvertes.fr/hal-00770023 Submitted on 4 Jan 2013
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Dynamic Substructuring of Damped Structures Using Singular Value Decomposition
R. Ohayon Chair of Mechanics, CNAM, 2 rue Conte,´ F-75003 Paris, France (mem. ASME)
R. Sampaio Department of Mechanical Engineering, Pontificia Universidade Catolica´ do Rio de Janeiro, 22453-900 Rio de Janeiro - RJ - Brazil
C. Soize Structures Department, ONERA, BP 72, F-92322 Chatillon Cedex, France Abstract
This paper deals with the theoretical aspects concerning linear elastodynamic of damped continuum medium in the frequency domain. Eigenvalue analysis and frequency response function are studied. The methods discussed here use a dynamic substructuring approach. The first method is based on a mixed variational formulation in which Lagrange multipliers are introduced to impose the linear constraints on the coupling interfaces. A modal reduction of each substructure is obtained using its free-interface modes. A practical construction of a unique solution is carried out using the Singular Value Decomposition (SVD) related only to the frequency-independent Lagrange multiplier terms. The second method is similar to the first one replacing the free-interface modes by the fixed-interface modes and elastostatic operator on the interface of each substructure.
1. Introduction field. As a consequence, the final system for the mixed formulation has a rank-deficiency in the matrix that describes the constraints. This leads In this paper, we are interested in eigenvalue and frequency response to non-uniqueness of the solution. In order to avoid this difficulty, a new function calculations of a linear dynamic three-dimensional bounded constructive approach is proposed consisting in using a Singular Value damped elastic structures subjected to prescribed forces. Recall that the Decomposition (SVD) of the frequency-independent constraint matrix frequency response functions allow deterministic and stationary random and chose a "least-square" solution that is in fact the solution of the orig- analyses to be performed (Kree and Soize, 1986; Argyris and Meljnek, inal problem. Due to a relatively small number of degrees of freedom 1991). More precisely, this paper is devoted to theoretical aspects of in the reduced model, the use of SVD is particularly efficient. Since the structure-structure coupling by dynamic substructuring methods using problem under consideration is linear, SVD is used only once. Conse- modal reduction procedures. The proposed methodology can be ap- quently, the SVD appears as an efficient and reliable tool to solve this plied to general linear coupled systems such as fluid-structure interaction rank-deficiency problem. It should be noted that SVD has been used for problems (Morand and Ohayon, 1995; Soize, Desanti and David, 1992). undamped linear vibration analysis of plates using dynamic substructur- For linear structural vibrations, dynamic substructuring techniques based ing by analytical methods (Jen, Johnson and Dubois, 1995). Let us recall on the use of the fixed-interface modes or free-interface modes (com- that SVD has also been used in the area of the nonlinear dynamical anal- pleted by static boundary functions, attachment modes, residual flexibil- ysis of multibody systems with nonlinear constraints (Singh and Likins, ity, etc.) of each substructure have been widely developed in the litter- 1985; Shabana , 1991; Schmidt and Muller,¨ 1993). ature: for conservative structures see for example (Hurty, 1965; Craig Now we give a short description of the content of each section. and Bampton, 1968; MacNeal, 1971; Rubin, 1975; Flashner, 1986; Min, Igusa and Achenbach, 1992; Farhat and Geradin, 1994) and for damped Section 2 deals with the displacement and mixed variational formulations structures (Klein and Dowell, 1974; Hale and Meirovitch, 1980; Leung, for the coupled linear structure-structure problem, Lagrange multiplier 1993; Farstad and Singh, 1995; Rook and Singh, 1995). field being introduced in the mixed problem. In Section 3, we present a dynamic substructuring method using the Some papers are based on a mixed formulation using a Lagrange multi- free-interface modes of each linear substructure. The modal reduction plier in order to impose the linear constraints on the coupling interfaces procedure is carried out using a new explicit construction of the La- (see Klein and Dowell, 1974; Min, Igusa and Achenbach, 1992; Farstad grange multiplier admissible space. Two practical constructions of the and Singh, 1995; Rook and Singh, 1995). Within the context of finite frequency response function of the global linear damped structure and element discretization of linear structural dynamic problems, Farhat and the eigenvalues of the associated conservative structure are performed Geradin (1994) have also introduced a Lagrange multiplier to take into using SVD once on a part of the linear system to be solved, namely on account incompatible meshes on the interface (their analysis is devoted the frequency-independent Lagrange multiplier terms. to undamped structures using a component mode method based on fixed- interface modes and static boundary functions). Section 4 is devoted to a dynamic substructuring method using the classi- cal Craig and Bampton fixed-interface modes and boundary static func- Below, we present an original general approach for damped structures tions of each linear substructure, presented in an original general frame- using continuum-based variational formulations and Ritz-Galerkin pro- work allowing various other decomposition procedures to be obtained. jection methods using free-interface modes and fixed-interface modes of After having constructed the reduced matrix model of each substructure, each substructure (in this paper we do not consider mathematical aspects we explain two procedures for the assemblage of the substructures and of error estimates connected to the truncation of the modal series). For the construction of a solution, (1) in a classical manner and (2) as in this purpose, various rigorous algebraic decompositions of admissible Section 3 using Lagrange multiplier field and SVD. classes of the unknown fields are introduced and leads to several linear Finally, in Section 5, some conclusions are presented. dynamic substructuring methods, the continuity of the displacement field on the interface being imposed through the use of a Lagrange multiplier
Journal of Applied Mechanics 1 Ohayon, Sampaio, Soize 2. Displacement and Mixed Variational For- sesquilinear forms on CΩr ×CΩr corresponding to the mass, stiffness and mulations for the Coupled Structure-Structure damping operators of substructure Ωr, are introduced as follows Problem mr(ur δur)= ρr ur δur dx (3) 2.1 General Mechanical Hypotheses Ωr In this section, the following hypotheses are introduced: r r r r r r - One considers the linear vibrations of a three-dimensional structure k (u δu )= σij (u ) εij ( δu ) dx (4) Ω about a static equilibrium configuration which is considered here as a r natural state (for the sake of brevity, prestress are not considered but r r r r r r d (u δu )= sij (u ) εij ( δu ) dx (5) could be added without changing the theory). Ωr - The structure is only submitted to prescribed external forces (no pre- It should be noted that the hermitian form mr is positive definite on scribed displacement). r r CΩr ×CΩr . The hermitian forms k and d are semi-definite positive With the above hypotheses, there are two cases. (degenerated forms) since rigid body displacement fields are allowed in (1)- The first one, which is the only case considered in this paper, cor- r 3 the present case. The set Rrig of Ê -valued rigid body displacement fields responds to prescribed external forces which are in equilibrium at each r (of dimension 6) is a subset of CΩr . Consequently, for all δu in CΩr , instant. Consequently, the displacement field of the structure is defined r r r r r r r r k (u δu ) and d (u δu ) are equal to zero for any u in Rrig. up to an additive rigid body displacement field. In this case, we are r We then define the following sesquilinear form z on CΩr ×CΩr only interested in the part of the displacement field due to the structural deformation. We will see below how the rigid body displacement field zr(ur δur)= −ω2 mr(ur δur)+iω dr(ur δur)+kr(ur δur) (6) can be disregarded. (2)- The second case corresponds to prescribed external forces which are Finally, we define fr by the relation not in equilibrium at some instants. To solve this problem, the method consists in transforming this case to the first case by adding an additional r r r r ≪f δu ≫= gΩr δu dx + gΓr δu ds (7) external force related to rigid body field. For the sake of brevity, this case Ωr Γr will not be considered in the present paper. 2.3 Continuum-Based Variational Formulations for Two Coupled One presents a variational formulation of the problem (first case), taking Substructures Ω1 and Ω2 into account an additional small structural damping based on a linear viscoelastic model with an instantaneous memory. A frequency do- We consider a structure composed of two substructures Ω1 and Ω2 that main formulation is used, the convention for the Fourier transform being interact through a common boundary Γ (the extension to the case of more u(ω)= e−iωt u(t) dt where ω denotes the circular frequency, u(ω) is than two substructures is straightforward). The notations introduced in
Ê 3
Ê C a vector in C and u(ω) its conjugate ( and denote the set of real and Section 2.2 are used with r =1 and r =2. The linear coupling conditions complex numbers respectively). on Γ are written as u1 = u2 on Γ (8) 2.2 Notation for a Substructure Ω r 1 1 2 2 σtot n = −σtot n on Γ (9) We consider a structure formed by substructures that will be denoted where nr is the unit normal to , external to r. by an index r. Let Ωr be the 3D-bounded domain occupied at static Γ Ω 1 2 equilibrium by the substructure labelled by index r. Let ∂Ωr = Γr ∪ Γ 2.3.1 Basic (u u ) Variational Formulation P0 with be the boundary of (assumed to be smooth). The Γr ∩ Γ = ∅ Ωr 1 2 1 2 boundary Γ will be the interaction surface with another substructure. The For all real ω in Ê and prescribed (f f ), find (u u ) in CΩ1 ×CΩ2 verifying the linear constraint u1 = u2 on Γ, such that, for all (δu1 δu2) external prescribed volumetric and surface force fields applied to Ωr and 1 2 r r r r in CΩ1 ×CΩ2 verifying the linear constraint δu = δu on Γ, one has Γr are denoted by gΩr and gΓr respectively. Let u =(u1 u2 u3) be the displacement field at each point x =(x1 x2 x3) in cartesian coordinates. 1 1 1 2 2 2 1 1 2 2
3 3 z (u δu )+ z (u δu )=≪f δu ≫ + ≪f δu ≫ (10) Ê The set of admissible displacement fields with values in C (resp. in ) is denoted by CΩr (resp. RΩr ) and is used for dissipative problems (resp. From the mathematical point of view (see Dautray and Lions, 1992), by associated conservative problems). For substructure Ωr, one denotes the 1 r 3 taking Sobolev space H C as admissible space , the existence r r (Ω ) CΩr test function (weighted function) associated with u as δu ∈CΩ (or in r and uniqueness of a solution of P0 can be proved. RΩr ) . The strain tensor is defined by 1 2
2.3.2 Mixed (u u Ð) Variational Formulation P1 1 ε (ur)= (ur + ur ) (1) This formulation consists in relaxing the linear constraint (defined by ij 2 i j j i Eq. (8)) used in P0 by the introduction of a Lagrange multiplier field Ð defined on Γ. Let ΛΓ be the admissible set of Lagrange multiplier fields in which v j denotes the partial derivative of v with respect to xj. The 3
defined on Γ with values in C . total stress tensor is defined by 1 2 1 2
Formulation P1 . For all real ω in Ê and prescribed (f f ), find (u u ) 1 2 r r r in CΩ1 ×CΩ2 and Ð in ΛΓ such that, for all (δu δu ) in CΩ1 ×CΩ2 and σtot = σ + iωs (2)
for all δ Ð in ΛΓ, one has r r r r
where σ is the elastic stress tensor defined by σ (u )= aijkh εkh(u ) 1 1 1 2 2 2 1 2 1 2 1 1 Ð ij z (u δu )+z (u δu )+b(Ð δu −δu )+b(δ u −u )=≪f δu ≫ + ≪f r r r and iωs is the viscous part of the total stress tensor such that sij(u )= (11) r bijkh εkh(u ) (using summation over repeated indices). The mechanical where b( ) is defined by coefficients aijkh and bijkh are independent of ω and verify the usual properties of symmetry and positivity (see Marsden and Hughes, 1983).
r r Ð r b(Ð u )= u ds (12) The mass density is denoted by ρ . For the dissipative problem, three Γ
Journal of Applied Mechanics 2 Ohayon, Sampaio, Soize Space of Traces on Γ. The set of the traces related to the boundary Γ, The present approach is based on the fact that any Ð in ΛΓ can be r r is denoted by CΓ. Therefore, if u ∈ CΩr , then the trace of u on Γ is expanded on a complete orthonormal set in CΓ and consequently, the r N denoted by u|Γ and belongs to CΓ. In Eq. (11), ΛΓ is the dual space of projection of the Lagrange multiplier Ð is done on the subspace WΓ N CΓ. of CΓ ⊂ ΛΓ. A characterization of WΓ requires the construction of a N basis of W denoted by {w1 w }. One possible method consists Remark. From the mathematical point of view (see Dautray and Li- Γ N
1 r 3 1 2 3 in extracting an independent system of N functions from the family C ons, 1992), by taking CΩr = H (Ω C )), CΓ = H (Γ ) and 1 1 2 2 N −1 2 3 {u1|Γ uN1 Γ u1|Γ uN2 Γ}. Consequently, for all Ð in WΓ , ΛΓ = H (Γ C ), the existence and uniqueness of a solution of | | one has formulation P1 can be proved using the so called LBB condition related N to the sesquilinear form b (see Brezzi and Fortin, 1991). It should be Ð = p w (18)
1 2 3 −1 2 3 γ γ C noted that H C is dense in H . (Γ ) (Γ ) γ =1
red 3. Dynamic Substructuring Using the Free-Inter- The Reduced Problem P1 . We use the Ritz-Galerkin method consist- face Modes of Each Substructure ing in substituting Eqs. (16) and (18) into Eq. (11). Using the orthogonal- ity conditions defined by Eqs. (14) and (15) and introducing the vectors The method is based on the use of the mixed variational formulation 1 1 1 2 2 2 of generalized coordinates q = (q1 qN1 ), q = (q1 qN2 ) and defined by P1. Then, a modal reduction is carried out using the Ritz- p = (p1 pN ), one deduces the following finite-dimension reduced Galerkin projection on the free-interface modes of each substructure. problem from P1 Finally, the Singular Value Decomposition (SVD) is used for the con- struction of the solution. 1 T 1 1 Z (ω) 0 B1 q F 2 T 2 F 2 3.1 Free-Interface Modes of a Substructure Ωr 0 Z (ω) B2 q = (19) B1 B2 0 p 0 A free-interface mode of a substructure Ωr (for r=1 or r=2) is defined as an eigenmode of the conservative problem associated with the substructure r 2 in which, for all real ω and for r = 1 and r = 2, [Z (ω)] is an (Nr × Ωr, subject to zero forces on Γ. The real eigenvalues ω ≥ 0 and the r Nr) complex symmetric matrix, [Br] a (N × Nr) real matrix which is eigenmodes u in RΩr are solutions of the following spectral problem: r Nr r independent of ω and F a C -valued vector. Matrix [Z (ω)] is defined find ω2 , ur ur 0 such that for all δur , one has ≥ 0 ∈RΩr ( = ) ∈RΩr by r 2 r r r 2 ω ω iω kr(ur δur)= ω mr(ur δur) (13) [Z ( )] = − [M ]+ [ D ]+[ K ] (20) r r r 2 where [M ] and [ K ] are diagonal positive-definite matrices such that It can be shown that there exist six zero eigenvalues 0=(ω−5) = [Mr] = r δ and [ Kr] = r ωr 2 δ , [ Dr] is a full symmetric ωr 2 (associated with the rigid body displacement fields) and αβ α αβ αβ α α αβ = ( 0) positive-definite matrix, such that [ Dr] = dr(ur ur ). Consequently, that the strictly positive eigenvalues (associated with the displacement αβ β α for all real ω, matrix [Zr(ω)] is invertible. Matrix [B ] is such that for field due to structural deformation) constitute the increasing sequence r r 2 r 2 r r all α in {1 Nr} and γ in {1 N}, one has 0 < (ω1) ≤ (ω2) . The six eigenvectors {u−5 u0} associated with zero eigenvalues span Rrig (space of the rigid body displacement r r r r [Br]γα = b(wγ uα) (21) fields). The family {u−5 u0; u1 } of all the eigenvectors forms a complete set in RΩ . For α and β in {−5 0; 1 }, we have the r Finally, vector F r is such that, for all α in {1 N }, one has orthogonality conditions r r r r r r r r Fα =≪f uα ≫ (22) m (uα uβ)= δαβ α (14)
r r r r r 2 k (u u )= δαβ ω (15) α β α α 3.3 Practical Construction of the Frequency Response Function of r red in which α > 0 is the generalized mass of mode α depending on the the Global Structure Using Reduced Problem P1 and SVD normalization of the eigenmodes. First, we introduce the (N × M) real matrix [ B ] such that 3.2 Modal Reduction of P1
Nr M = N1 +N2 [ B ]=[ B1 B2] (23) We introduce the subspace CΩr of CΩr , of dimension Nr, spanned by r r r Nr {u1 u } with Nr ≥ 1. For all u in CΩ , one has Nr r and write Eq. (19) as
Nr ur qr ur Z(ω) BT q F = α α (16) = (24) α =1 B 0 p 0
r in which qα are complex-valued generalized coordinates. Concerning the In order to solve Eq. (24), we use a Singular Value Decomposition trace of the displacement field (including rigid body displacement field) (SVD) of [ B ]. It is know that there exist algorithms (see Golub and Van on , the subspace spanned by the family ur ur ur is Γ { −5|Γ 0|Γ; 1|Γ } Loan, 1989) which are very efficient for the construction of the SVD a complete set in CΓ (for the two domains r =1 and r =2). Consequently, of reasonable size matrices. This is the case for the reduced problems r the family {u1|Γ } forms a complete set of the displacement field on obtained by modal projection as Eq. (24). In the proposed approach, Nr Γ due only to the structural deformation. Let CΓ be the subspace of CΓ it should be noted that SVD will only be applied to the submatrix [ B ] spanned by the finite family {ur ur }. Let WN be the subspace in Eq. (24). The SVD of (N × M) real matrix [ B ] with M ≥ N (see 1|Γ Nr |Γ Γ of CΓ of finite dimension N ≤ N1 + N2 defined by Section 3.2) consists in constructing the following decomposition
N N1 N2 T WΓ = CΓ ∪CΓ (17) [ B ]=[U ][Σ][V ] (25)
Journal of Applied Mechanics 3 Ohayon, Sampaio, Soize −1 k where [U ] is an (N × N) orthogonal real matrix, [V ] is an (M × M) ek =< [ Z(ω)] F V >. Then, the projection of Eq. (33) on the orthogonal real matrix and [Σ] is a (N× M) real matrix which is written remaining {Vn+1 VM } yields for all k in {n+1 M}, in block form as [Σ]= Σ+ 0 (26) n −1 k′ k −1 k ξ = − y ′ <[ Z(ω)] V V > + <[ Z(ω)] F V > + k k in which [0] is the (N× (M − N)) null matrix and [ Σ ] is the (N× N) k ′=1 diagonal matrix of positive or null singular values σk such that σ1 ≥ (35) σ2 ≥ ≥ σN ≥ 0. Let n be the integer such that 1 ≤ n ≤ N such that The corresponding algorithm is summarized below. Step 0: calculating the SVD of [ B ] in order to obtain its rank n and σ1 ≥ σ2 ≥ ≥ σn > σn+1 = = σN = 0 (27) V1 VM . Consequently, the rank of [ B ] is equal to n and Eq. (25) yields the SVD Then, for each real ω, expansion Step 1: solving the linear equation of dimension n with n+1 right-hand n side members {F; V1 Vn} k kT [ B ]= σk U V (28) k =1 [ Z(ω)] X0 = F ; [ Z(ω)] Xk = Vk k ∈ {1 n} ; (36) in which the vectors Uk and Vk are the columns of [U ] and [V ] and such that Step 2: constructing (n× n) complex symmetric matrix [ E(ω)] such j k j k k k′ ′ = δjk
solution q of Eq. (24) in the null space of [ B ], i.e. [ G(ω)] Ü = g (38)
M M−n C k in which Ü =(ξn+1 ξM ) is a vector in , g =(g1 gM−n) q = ξk V (31) M−n k+n is a vector in C such that gk =< F V > and [ G(ω)] is a k= n+1 ((M − n) × (M −n)) complex symmetric matrix such that Using Eqs. (28) and (29), it can be seen that q defined by Eq. (31) satisfies Eq. (30). Using Eqs. (28) and (31), Eq. (24) yields [ G(ω)]= −ω2 [ M ]+ iω [ D ]+[ K ] (39)
M n where [ M ], [ D ] and [ K ] are ((M − n) × (M − n)) real symmetric ξ [ Z(ω)] Vk + σ η Vk = F (32) k k k positive-definite matrices defined, for all k and k′ in {1 M −n}, by k= n+1 k =1
k+n k′+n k ′ < V V > in which ηk =, or equivalently, [ M ]k k = [ M ] (40) M n k+n k′+n k −1 k −1 [ D ]k′k =<[ D ] V V > (41) ξk V + σk ηk [ Z(ω)] V =[ Z(ω)] F (33) k= n+1 k =1 k+n k′+n [ K ]k′k =<[ K ] V V > (42)
Equation (32) or (33) shows that ξk can be calculated in a unique way. The corresponding algorithm is summarized below. 3.3.2 Second Algebraic Stage of the Practical Construction of Solu- Step 0: calculating the SVD of [ B ] in order to obtain its rank n and tion. Vn+1 VM . First Procedure. Then, for each real ω, The projection of Eq. (33) on {V1 Vn} yields Step 1: constructing ((M−n)×(M−n)) complex symmetric matrix [ G ] such that k+n k′+n ′ [ E(ω)] y = e (34) [ G ]k′k =<[ Z(ω)] V V > for k and k in {1 M −n}; M−n k+n Step 2: constructing C -valued vector g such that gk =
Journal of Applied Mechanics 4 Ohayon, Sampaio, Soize Comments on the two proposed procedures. 4.1 Reduced Matrix Model of Substructure Ωr (1)- Due to the fact that we have to solve a reduced size problem N and r M are small. 4.1.1 Basic u Variational Formulation for Substructure Ωr
(2)- In the first procedure, Step 1 is solved substructure by substructure Consider substructure Ωr submitted to the external applied forces gΩr in independently. For each substructure Ωr, if the damping operator defined Ωr, gΓr on Γr and gΓ on the interaction surface Γ. by Eq. (5) is diagonalized by the free-interface modes of this substruc- The basic variational formulation for substructure Ωr is written as fol- ture, Step 1 is straightforward. If not, we have to solve a small (Nr×Nr) lows. r r full complex symmetric system for each substructure. In Step 4, one Basic problem P1 . For all real ω in Ê and prescribed f defined by Eq. r r has to solve a linear system of dimension n with a full (n× n) com- (7), find u in CΩr such that, for all δu in CΩr , one has plex symmetric matrix corresponding to the total number of independent r r r r linear constraints existing in the global structure (assemblage of all the z (u δu )=≪f δur ≫ + ≪fΓ δur ≫ (46) substructures). (3)- In the second procedure, Step1 is relative to the global structure r r r in which z is defined by Eq. (6) and where ≪fΓ δu ≫= Γ gΓ δu ds. (assemblage of all the substructures) and Step 4 requires to solve a full complex symmetric linear system of dimension M −n. 4.1.2 Fixed-Interface Modes of Substructure Ωr (4)- For example, if there are NS substructures (in this paper NS = 2) A fixed-interface mode of a substructure Ωr (for r=1 or r=2) is defined and if the mean value of {Nr} on the set of substructures is NR = as an eigenmode of the conservative problem associated with the sub- 1 NS 3 N , the order of floating operations is N N for the first structure Ωr, which is fixed on Γ. Since the problem is conservative and NS r=1 r S × R procedure with a damping matrix of each substructure which is not diag- defined in a bounded domain, all the quantities are real. Consequently, 3 4 0 onalized by the free-interface modes of this substructure and, NS × NR we introduce the set RΩr defined by for the second procedure. 0 r r As a conclusion, the first procedure is recommended since it is more RΩr = δu ∈RΩr δu = 0 on Γ (47) efficient (particularly, if the damping matrix of each substructure is diag- 2 onalized by the free-interface modes of this substructure). in which RΩr is defined in Section 2.2. The real eigenvalues ω > 0 and the eigenmodes ur in R0 are solution of the following spectral problem: 3.4 Practical Construction of the Eigenmodes of the Global Structure Ωr Find ω2 > 0, ur ∈R0 (ur = 0) such that for all δur ∈R0 , one has Using a Reduced Spectral Problem and SVD Ωr Ωr
The conservative problem associated to Eq. (24) leads to the following kr(ur δur)= ω2 mr(ur δur) (48) spectral problem
1 T 1 1 1 in which mr and kr are defined by Eqs. (3) and (4) respectively. It K 0 B1 q M 0 0 q 2 T 2 2 2 2 can be shown that the eigenvalues constitute and increasing sequence 0 K B2 q = ω 0 M 0 q (43) r 2 r 2 r r B1 B2 0 p 0 0 0 p 0 < (ω1) ≤ (ω2) . The family {u1 u2 } of the eigenvectors 0 associated with the eigenvalues, forms a complete set in RΩr . For α and in which the two matrices defined by blocks are real symmetric and β in {1 2 }, we have the orthogonality conditions similar to Eqs. (14) independent of ω. Using a global notation as done in Eq. (24), Eq. (43) and (15). is rewritten as 4.1.3 Introduction of the Elastostatic Lifting Operator Sr T K B q 2 M 0 q r = ω (44) We consider the solution ustat of the elastostatic problem of substructure B 0 p 0 0 p r Ωr subjected to a prescribed displacement field u|Γ on Γ. Let RΓ and ur For this problem, we must use the second procedure defined in Section R |Γ be the sets of functions such that 3.3.2 (in this case, the first procedure cannot be directly used since Ωr [K] − ω2 [M] is not invertible for all real values of ω). Substituting Eq. RΓ = { x → uΓ(x) ∀ x ∈ Γ } (49) (31) in the first row of Eq. (44), projecting it on {Vn+1 VM } and using Eq. (29), yield r u |Γ r r r R = u ∈RΩ u = u on Γ (50)
2 Ωr r |Γ Ü [ K ] Ü = ω [ M ] (45) r The field ustat satisfies the following variational formulation in which [ M ] and [ K ] are defined by Eqs. (40) and (42). The corresponding algorithm is summarized below. ur kr(ur δur) = 0 ur ∈R |Γ ∀ δur ∈R0 (51) Step 0: Calculating the SVD of [ B ] in order to obtain its rank n and stat stat Ωr Ωr Vn+1 VM . ur Step 1: constructing ((M− n) × (M−n)) real symmetric matrices [ M ] 0 |Γ r r where RΩr is the space RΩr obtained for u|Γ = 0. The solution ustat of r and [ K ]; Eq. (51) defines the linear operator S from RΓ into RΩr (called lifting Step 2: solving the generalized eigenvalue problem defined by Eq. (45); operator in mathematics), such that Step 3: calculating the eigenmodes u = (u1 u2) of the structure by r r r r using Eqs. (31) and (16). u|Γ → ustat = S (u|Γ) (52)
4. Dynamic Substructuring Using the Fixed- r Γ We denote the range space of operator S as RΩr ⊂ RΩr such that Interface Modes of Each Substructure Γ r r RΩr = S (RΓ). It should be noted that the discretization of S by the In this section, we present a modal reduction procedure based on formu- finite element method is obtained by a classical static condensation pro- lation P1 using SVD (see Section 2.3.2) starting from a reduced matrix cedure (sometimes called the Schur complement) of the stiffness matrix model for each substructure Ωr. of substructure Ωr with respect to degrees of freedom on Γ.
Journal of Applied Mechanics 5 Ohayon, Sampaio, Soize r r r r r 4.1.4 Conjugate Relationships Between uα and ustat in which q = (q1 qNr ) is the vector of generalized coordinates r r r r related to the fixed-interface modes, F = (F1 FNr ) is the vector Taking δu = uα in Eq. (51), for ustat satisfying Eq. (51) yields whose components are given by Eq. (22) using the fixed-interface modes kr(ur ur ) = 0 (53) and fΓ is defined in Section 4.1.1. stat α r (1)- For all real ω, linear operator ZΓ(ω) is defined by the following Γ Γ r r 0 r sesquilinear form on CΩr ×CΩr For a given mode (ωα uα ∈ RΩr ), the modal reaction forces Fα = σr(ur ) nr on Γ is defined by the variational property α r r r r r r r r ≪ZΓ(ω) u|Γ δu|Γ ≫= z (S (u|Γ) S (δu|Γ)) (65) r r r r 2 r r r r r r k (uα δu ) − (ωα) m (uα δu )= Fα δu ds ∀ δu ∈RΩr From Eq. (6), we deduce the following abstract operator equation Γ (54) r 2 r r r ZΓ(ω)= −ω MΓ + iω DΓ + KΓ (66) Using Eqs. (48) and (53), Eq. (54) yields r r r in which the mass, damping and stiffness operators MΓ, DΓ and KΓ are r r r 1 r r defined by m (ustat uα)= − r 2 Fα u|Γ ds (55) (ωα) Γ r r r r r r r r ≪MΓ u|Γ δu|Γ ≫= m (S (u|Γ) S (δu|Γ)) (67) r r 0 Consequently, for all field u|Γ in RΓ and uα in RΩr , one has r r r r r r r r ≪DΓ u|Γ δu|Γ ≫= d (S (u|Γ) S (δu|Γ)) (68) r r r r k (S (u|Γ) uα) = 0 (56) r r r r r r r r ≪KΓ u|Γ δu|Γ ≫= k (S (u|Γ) S (δu|Γ)) (69) r r r r r r r 1 r r where m , k and d are defined by Eqs. (3), (4) and (5), respectively. m (S (u|Γ) uα)= − r 2 Fα u|Γ ds (57) (ωα) Γ It should be noted that these operators are related to surface Γ and cor- respond to the static condensation on Γ of the mass, stiffness (Guyan, r 4.1.5 Decomposition of RΩr and CΩr 1965) and damping operators using the elastostatic operator S defined r r in Section 4.1.3. Due to the fact that the trace of u − ustat is zero on Γ, we have the r following decomposition (2)- For all real ω, the (Nr × Nr) complex symmetric matrix [Z (ω)] is defined by Eq. (20) using the fixed-interface modes. It should be noted Γ 0 that if the damping operator defined by Eq. (5) is diagonalized by the RΩr = RΩr ⊕RΩr (58) fixed-interface modes, matrix [Zr(ω)] is diagonal. ∞ (3)- For all real ω, the linear operator Ar(ω) is defined by the following r r r r r Γ Nr
u = S (u )+ q u (59) C |Γ α α sesquilinear form on CΩr × α =1 Γ 0 Γ 0 Nr Let C and C be the complexified vector spaces of R and R r r r r r r Ωr Ωr Ωr Ωr ≪A (ω) u δqr ≫= z (S (u ) u ) δqr (70) respectively. One then has |Γ |Γ α α α =1
Γ 0 r r r CΩr = CΩr ⊕CΩr (60) in which δq =(δq1 δqNr ). From Eq. (6), we deduce the following abstract operator equation r 3 r and Eq. (59) holds with u being a C -valued field and q complex |Γ α Ar 2 Ar Ar numbers. (ω)= −ω m + iω d (71) Ar Ar 4.1.6 Construction of the Reduced Matrix Model in which m and d are operators defined by We introduce the subspace C0 Nr of C0 , of dimension N , spanned by Ωr Ωr r Nr r r Γ Nr {u u } with N ≥ 1 and the subspace C of CΩ such that Ar r r r r r r r 1 Nr r Ωr r ≪ m u|Γ δq ≫= m (S (u|Γ) uα) δqα (72) α =1 CΓ Nr = CΓ ⊕C0 Nr (61) Ωr Ωr Ωr Nr r r r r r r r r ≪A u Γ δq ≫= d (S (u Γ) u ) δq (73) r r Nr d | | α α For all u and δu in CΩr , one has α =1 in which Eq. (56) has been used. The quantities mr(Sr(ur ) ur ) are Nr |Γ α r r r r r r r r r calculated using Eq. (57) and d (S (u Γ) uα) using Eqs. (5), (51) and u = S (u Γ)+ qα uα (62) | | t r α =1 (52). Finally, operator A (ω) is defined by the following sesquilinear Nr Γ form on C ×CΩr such that Nr r r r r r Nr δu = S (δu|Γ)+ δqα uα (63) tAr r r r r r r r α =1 ≪ (ω) q δu|Γ ≫= qα z (uα S (δu|Γ)) (74) 0 α =1 We use the Ritz-Galerkin method related to space CΩr consisting in substituting Eqs. (62) and (63) into Eq. (46). Using the conjugate In conclusion, the matrix (of operators) in the left-hand side of Eq. (64) relations (56) and (57) and the orthogonality properties (14) and (15) for is called the "reduced matrix model" of substructure Ωr relative to the r fixed-interface modes, we obtain in abstract operator notation displacement field u|Γ on Γ and the Nr generalized coordinates (which can be viewed as "internal generalized degrees of freedom"). We refer r t r r Z (ω) A (ω) u fΓ to Morand and Ohayon (1995) for the particular case of an undamped Γ Γ = (64) Ar(ω) [Zr(ω)] qr F r structure.
Journal of Applied Mechanics 6 Ohayon, Sampaio, Soize 4.2 Frequency Response Function and Eigenmodes Constructions 5. Conclusion for the Global Structure Using the Mixed Variational Formulation Within a general continuum-based approach, we have presented two dy- and SVD namic substructuring procedures by modal reduction methods in order 4.2.1 Modal Reduction of Mixed Problem P1 to calculate the frequency response function of linear damped struc- tures and the eigenmodes of the associated conservative systems. The The reduction of P1 defined in Section 2.3.2 is obtained using the reduced free-interface and fixed-interface modes of each substructure are used matrix model defined by Eq. (64) for each substructure. Recall that the within a mixed variational formulation involving Lagrange multiplier N projection of Lagrange multiplier Ð must be done on the subspace WΓ of fields defined on the coupling interfaces. Generally, the introduction of N CΓ ⊂ ΛΓ. A characterization of WΓ requires the construction of a basis a Lagrange multiplier field associated with kinematic linear constraints N N of WΓ denoted by {w1 wN }. Consequently, for all Ð in WΓ , one induces some difficulties for the construction of the solution due to the N has Eq. (18). Substituting Eqs. (62),(63), (18) and δ Ð = γ=1 δpγ wγ rank deficiency of the obtained linear system. In the present paper, the into Eq. (11), we obtain Singular Value Decomposition (SVD) method is applied to the frequency- independent Lagrange multiplier terms. The use of SVD is particularly 1 tA1 tB 1 ZΓ(ω) (ω) 0 0 1 uΓ 0 efficient due to a relatively small number of degrees of freedom in the A1 1 1 F 1 (ω) [Z (ω)] 0 0 0 q reduced model and is used once. Therefore, the SVD appears as an 2 tA2 tB 2 0 0 ZΓ(ω) (ω) 2 uΓ = 0 efficient and reliable tool for this problem. A2 2 2 F 2 0 0 (ω) [Z (ω)] 0 q B B 1 0 2 0 0 p 0 References (75) in which we can recognize the reduced model of each substructure (see Argyris, J., and Meljnek, H.P., 1991, Dynamics of Structures, Nort- Eq. (64)). Using Eq. (12), for r = 1 2 and γ in {1 N}, operators Holland, Amsterdam. B1 and B2 are defined by Brezzi, F., and Fortin, M., 1991, Mixed and Hybrid Finite Element Methods, Springer. r [Br]γ = b(wγ uΓ) (76) Craig Jr., R.R., 1985, "A Review of Time Domain and Frequency Do- main Component Mode Synthesis Method,", Combined Experimental- Analytical Modeling of Dynamic Structural Systems, edited by Martinez, 4.2.2 Practical Construction of the Frequency Response Function D.R. and Miller, A.K., ASME-AMD, Vol. 67. Using SVD Craig Jr., R.R., and Bampton, M.C.C., 1968, "Coupling of Substruc-
Since B1 and B2 are independent of ω, Eq. (75) can be rewritten as tures for Dynamic Analysis," AIAA Journal, Vol. 6., pp. 1313-1319. Dautray R., and Lions J.-L., 1992, Mathematical Analysis and Numer- Z(ω) tB Q F ical Methods for Science and Technology, Springer-Verlag, Berlin. = (77) B 0 p 0 Farhat, C., and Geradin, M., 1994, "On a Component Mode Method and its Application to Incompatible Substructures," Computer and Struc- 1 1 2 2 tures, Vol. 51, No. 5, pp. 459-473. where Q = (uΓ q uΓ q ). Equation (77) being similar to Eq. (24), the practical construction is carried out as described in Section 3.3. Farstad, J.E., and Singh, R., 1995, "Structurally Transmitted Dynamic Power in Discretely Joined Damped Component Assemblies", J. Acoust. 4.2.3 Practical Construction of the Eigenmodes Using SVD Soc. Am., Vol. 97, No. 5, pp. 2855-2865. The conservative problem associated to Eq. (75) leads to the following Flashner, H., 1986, "An Orthogonal Decomposition Approach to Modal spectral problem Synthesis", Int. J. Num. Meth. Eng., Vol. 23, No. 3, pp. 471-493. Golub, G.H., and Van Loan, C.F., 1989, Matrix Computations, 2nd 1 t 1 Edition, The John Hopkins University Press, Baltimore and London. KΓ 0 0 0 B1 uΓ 1 1 Guyan, R.J., 1965, Reduction of Stiffness and Mass Matrices, AIAA 0 [K ] 0 0 0 q 2 t 2 Journal, Vol. 3. 0 0 KΓ 0 B2 uΓ 0 0 0 [K2] 0 q2 Hale, A.L., and Meirovitch, L., 1980, " A General Substructure Synthe- sis Method for the Dynamic Simulation of Complex Structures", Journal B1 0 B2 0 0 p of Sound and Vibration, Vol. 69, No. 2, pp. 309-326. 1 tA1 1 Hurty, W.C., 1965, "Dynamic Analysis of Structural Systems Using MΓ m 0 0 0 uΓ A1 1 1 Component Modes," AIAA Journal, Vol. 3, No. 4, pp. 678-685. m [M ] 0 0 0 q 2 2 tA2 2 Jen, C.W., and Johnson, D.A., and Dubois, F., 1995, "Numerical = ω 0 0 MΓ m 0 uΓ (78) A2 2 2 Modal Analysis of Structures Based on a Revised Substructure Synthesis 0 0 m [M ] 0 q 0 0 0 0 0 p Approach", Journal of Sound and Vibration, Vol. 180, No. 2, pp. 185- 203. Equation (78) is rewritten using the global notation introduced in Eq. (77) Klein, L.R., and Dowell, E.H., 1974, "Analysis of Modal damping and is then similar to Eq. (44). Consequently, we can use the method by Component Modes Method Using Lagrange Multipliers", Journal of presented in Section 3.4 for solving this spectral problem. Applied Mechanics, Vol. 39, pp.727-732. 4.2.4 General comments Kree, P., and Soize, C., 1986, Mathematics of Random Phenomena, Reidel, Dordrecht. In the case of a finite element discretization with incompatible mesh on Leung, A.Y.T., 1993, Dynamic Stiffness and Substructures, Springer- Γ, the method presented in Section 4.3 (Eqs. (75) and (78)) is efficient Verlag, New York. because, since B1 and B2 are independent of ω, the SVD is carried MacNeal, R.H., 1971, "A Hybrid Method of Component Mode Synthe- out once and for all (even if the sizes of the matrices of the discretized sis," Computers and Structures, Vol. 1, pp. 581-601. operators B1 and B2 are important). Marsden J.E., and Hughes T.J.R., 1983, Mathematical Foundations of Elasticity, Prentice Hall, Englewood Cliffs, New Jersey.
Journal of Applied Mechanics 7 Ohayon, Sampaio, Soize Min, K.W., Igusa, T., and Achenbach, J.D., 1992, "Frequency Window for Multibody Systems with Constraints," Advanced Multibody System Method for Strongly Coupled and Multiply Connected Structural Sys- Dynamics, Edited by W. Schiehlen, Kluwer Academic Publishers, Dor- tems: Multiple-Mode Windows", Journal of Applied Mechanics, Vol. drecht/Boston/London, pp. 427-433. 59, pp.244-252. Shabana, A.A., 1991, "Constrained Motion of Deformable Bodies," Int. Morand, H.J.P., and Ohayon, R.,1995, Fluid Structure Interaction, J. Num. Meth. Eng., Vol. 32, pp. 1813-1831. Wiley, New York. Rubin, S., 1975, "Improved Component Mode Representation for Struc- Singh, R.P., and Likins, P.W., 1985, "Singular Value Decomposition for tural Dynamic Analysis," AIAA Journal, Vol. 18, No 8, pp. 995-1006. Constrained Dynamical Systems," Journal of Applied Mechanics, Vol. Rook, T.E., and Singh, R., 1995, "Power Flow Through Multidimen- 52, pp. 943-948. sional Compliant Joints Using Mobility and Modal Approaches", J. Soize, C., Desanti, A., and David, J.M., 1992, "Numerical Methods in Acoust. Soc. Am., Vol. 97, No. 5, pp. 2882-2891. Elastoacoustic for Low and Medium Frequency Ranges," La Recherche Schmidt, Th., and Muller,¨ P.C., 1993, "A Parameter Estimation Method Aerospatiale´ (English Edition), Vol. 5, pp. 25-44.
Journal of Applied Mechanics 8 Ohayon, Sampaio, Soize