Lifetime-Oriented Structural Design Concepts- P19 docx

498 4 Methodological Implementation u [ mm ] F [ kN ] 0.3 5 0.30.250.20.150.10.050 350 300 250 200 150 100 50 0 Fig. 4.94. Numerical investigation of crack propagation of an anchor pull-out test: Load-displacement curve displacement reaches u z ≈ 0.21 [mm] the carck propagation nearly stops due to the fact that most of the load is carried by the area of counter pressure. This is evident from Figure(4.93,right) that shows the strongly increased value of the stress component σ 33 at the counter pressure support. On the contrary in the beginning of the cracking process (figure(4.93,left)) there is nearly no compression at the area of counter pressure because of the load carrying of the concrete in the vicinity of the anchor plate. When the displacement reaches u z ≈ 0.21 [mm], the load is primarily induced to the counter pressure sup- ports by the nearly fully separated upper part of the concrete structure. This bearing behaviour of the system is primarly characterized by the position of the crack surface that is running underneath the pressure supports. In the case that the crack reaches the upper surface of the concrete structure being in front of the supports, there is going to be a structural softening of the concrete block. This interpretation is similiary to the one in [301] where a similiar numerical test is performed. As can be further deduced from Figure 4.93 there is a high compression atop of the anchor plate, which would also lead to compression failure of concrete. This does not occur here, because this analysis only takes failure of concrete in tension into account. 4.2.10 Substructuring and Model Reduction of Partially Damaged Structures Authored by Christian Rickelt and Stefanie Reese The objective of this section is to present an efficient strategy to cope with demanding dynamical simulations of complex structures. We are in par- ticular interested in long term calculations like life cycle investigations. Our 4.2 Numerical Methods 499 ansatz emanates from the idea that a structure only comprises locally dis- tributed damaged zones. The proposed method is an evolution of the classical Craig-Bampton approach to partially damaged problems and incorporates an advantageous decomposition of the entire structure into linear and non- linear segments. The former are reduced by model reduction techniques. In the nonlinear components the deterioration of the material is simulated by a continuum damage model for ductile damage phenomena of metals. This material model is included into a new solid beam finite element formulation Q1SPb based on reduced integration with hourglass stabilisation. The section closes with an example to validate the proposed strategy and to determine the approximation error caused by the model reduction of the linear components. 4.2.10.1 Motivation and Overview The importance of simulation tools for the design and lifetime estimation of complex structures increases noticeably. Highly demanding tasks and re- quirements on the accuracy of numerical computations necessitate powerful simulation techniques and more elaborate models. Since in general these mod- els cannot be solved analytically the underlying ordinary or partial differential equations are often discretised and solved by the finite element method. Dynamic problems require beside the spatial discretisation as well a dis- cretisation in time. This can be accomplished by implicit or explicit time integration schemes. While the former lead to large, sparse equation systems, whose solution is memory and time consuming, the disadvantage of the ex- plicit methods is that they are only conditionally stable. Accordingly its time step size is limited by a critical time step. As a consequence the latter schemes are not suitable for long-term computations. Strategies to solve such complex long-term calculation numerically efficient have been a topic of research in the field of engineering technologies and math- ematics. More related to the former one are the condensation methods and the staggered schemes. The domain decomposition methods are mostly researched in mathematics while the model reduction techniques are established in both communities. The idea of model reduction is to transform the large equation system into a small, dynamically equivalent substitute model. Powerful model reduc- tion methods exist for the solution of linear and weakly nonlinear systems. Highly nonlinear problems like the evolution of damage cannot be solved. [761] and [808] employ nonlinear model reduction techniques to reduce geometri- cally nonlinear problems and fluid-structure interactions, respectively. For an overview of model reduction methods in general refer to the publications of [111, 761, 49, 50, 651, 288, 76, 51, 451, 287, 236, 583, 485, 813]. Linear model reduction methods can be classified into three different groups: the singular value decomposition (SVD)-based methods, the Krylov- based methods and the condensation methods. Out of the group of SVD- based model reduction two well-known methods are the Modal Reduction 500 4 Methodological Implementation (MODRED) (see the standard text books [195, 203]) and the Proper Orthog- onal Decomposition (POD) (see [121, 384, 755]). Two important members of the Krylov-based methods are the Load-Dependent Ritz Vectors (LDRV) (see [328, 203, 585]) as well as the Pad ´ e-Via-Lanczos algorithm (PVL). In this publication a symmetric version for undamped second order systems, called symmPVL, is used (see e.g. [288, 76]). The above mentioned group of condensation methods comprise model re- duction methods which condense the inner degrees-of-freedom (dofs) and con- serve the physical interface dofs. This idea can be structured into static as well as dynamic condensation methods. According to [620] the latter yield exactly reduced dynamical systems which even for linear problems result in nonlinear functions which depend on the frequency. Hence they are not numerically effi- cient. In contrast the static condensation of [333] is suitable to only a limited extent for the model reduction of linear dynamic systems. Corresponding to the text book of [651] component mode synthesis techniques (CMS techniques) establish a significant extension of Guyan’s reduction to methods with hy- brid transformation matrices. Initialised by the classical papers of [401] and [216] (see also the review article of [215]), CMS techniques with fixed inter- faces have been developed within the scope of reduction methods for large structural dynamic models and the design of single dynamical components of challenging structures. These methods are based on the superposition of different contribution of the deformations. Hence they are only valid for lin- ear problems. For further information on CMS techniques see the overview publications of [217, 206, 498, 722, 511]. Alternatively the objective of staggered algorithms (see for example [548]) is to solve multi-field or dynamic problems which are discretised by different time integration schemes (implicit/explicit) and varying time step lengths (subcycling) within each subdomain (see the classical papers of [108] and [107]). An interesting approach of an explicit-implicit multi time step method for nonlinear structural dynamics which prescribes the continuity of velocities at the interface and uses a dual Schur formulation has been published by [323]. [274] and [275] extend this ansatz successively to nonmatching meshes and linear as well as nonlinear model reduction techniques. The primary interest of domain decomposition methods, grouped into over- lapping (”Schwarz methods”) and non-overlapping (”iterative substructur- ing methods”) methods, is the development of highly efficient parallelised iterative solution techniques. These usually result in conjugate gradient schemes. The latter are subdivided into the Neumann-Neumann-(Bal- anced Domain Decomposition (BDD), [515]), Dirichlet-Neumann-and Dirichlet-Dirichlet-Algorithm (Finite Element Tearing and Interconnect- ing Method (FETI), [273]). Most of the publications deal with linear problems. Applications to geometrically nonlinear problems can be found in the papers of [218] and [272]. 4.2 Numerical Methods 501 Structural Dynamics Modelling (FEM) Linear Components Nonlinear Components Model Reduction Substructure Technique Damage Modelling Fig. 4.95. Concept for the efficient simulation of dynamic, partially damaged struc- tures by means of model reduction and substructuring 4.2.10.2 Concept In this section an efficient strategy for dynamic long-term simulations of com- plex structures with local nonlinearities by means of the finite element method is presented. Our approach emanates from the idea that engineering structures are usually designed in such a way that most components of a structure be- have linearly elastically and under the assumption of small deformations. Thus undesired effects such as the evolution of damage or possibly occurring large deformations are localised in small parts of a structure. This ansatz, depicted in Figure 4.95, enables to decompose any discretised structure strictly into its linear and nonlinear components. In the latter the evolution of ductile damage of metals is considered. The damage model is based on the void growth model of [691, 690]. Further we assume that the evolution of damage is only influenced indirectly by the dominating linear subsystems. Hence, to increase the efficiency of the strategy, the latter are re- duced by model reduction methods in conjunction with the Craig-Bampton method – one of the widely used CMS substructure techniques. Beside the common Modal Reduction, linear model reduction methods of superior accu- racy – the Pad ´ e-Via-Lanczos algorithm (PVL), the Load-Dependent Ritz Vectors (LDRV) and the Proper Orthogonal Decomposition (POD) – are em- ployed. Finally the substructuring of the total structure into reduced lin- ear and nonlinear components is exploited. Instead of solving a large mono- lithic equation system the system response of the redundant interfaces of the components is computed. Subsequently the inner dofs of all components are calculated. One advantage of the presented concept rests upon the beneficial combi- nation of well-known and robust linear model reduction methods, the CMS 502 4 Methodological Implementation technique, the material as well as the efficient element formulation. Another important aspect is the numerical implementation. For this purpose the math- ematical development environment Matlab and the finite element program Feap are linked by the interface Feapmex 1 . 4.2.10.3 Derivation of a Substructure Technique for Nonlinear Dynamics In this section the Craig-Bampton method is summarised. Afterwards the linear model reduction methods and the employed substructure technique are discussed. 4.2.10.3.1 Craig-Bampton Method The concept of CMS techniques has been developed [401] and later rewritten and simplified by [216] to analyse complex structural systems decomposed into interconnected components with fixed interfaces. The Craig-Bampton method superposes two different fractions of the motion: the static or con- straint modes Ψ ic of the matrix Ψ T :=  Ψ T ic I T cc  are defined as the static deformation of a structure when a unit displacement is applied to one in- terface dof while the remaining interface dofs are restrained. The matrix I cc herein is an identity matrix of dimension c × c. The indices i and c indicate the inner and the interface dofs, respectively. The second fraction are the k (k  i) remaining inner dynamical or so-called normal modes φ j ,j=1, ,k of the fixed subsystem s. They are stored column by column into the matrix Φ r := ⎡ ⎢ ⎣ φ 1 , ,φ k O ck ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ Φ ik O ck ⎤ ⎥ ⎦ . (4.243) The matrix O ck is a zero matrix of dimension c × k. Usually the modes Φ ik are represented by eigenvectors. In this contribution the LDRV-, POD- or symmPVL-vectors of alternative model reduction methods of superior accu- racy, as presented in section 4.2.10.3.2, are utilised. With these modes the physical coordinates u can be transformed by the relation ⎡ ⎢ ⎣ u i u c ⎤ ⎥ ⎦    u (s) = ⎡ ⎢ ⎣ Φ ik Ψ ic O ck I cc ⎤ ⎥ ⎦    V (s) CB ⎡ ⎢ ⎣ q k q c ⎤ ⎥ ⎦    q (s) (4.244) into generalised coordinates q. V CB symbolises the well-known Craig-Bamp- ton transformation matrix of one single component s. Transforming with 1 For further information see the following homepage: www.cims.nyu.edu/ dbindel/feapmex/feapmex/doc/feapmex.html 4.2 Numerical Methods 503 the relation (4.244) the linear equation of motion of one component s by an orthogonal projection V (s) CB T M (s) V (s) CB    M (s) rr ¨ q (s) (t)+V (s) CB T K (s) V (s) CB    K (s) rr q (s) (t)=V (s) CB T b (s)    b (s) r f(t) (s) (4.245) leads for linear undamped systems to a component of smaller dimension in which the interface dofs u c = q c are conserved in physical coordinates. In the latter equation M, K, b and f(t) are the mass matrix, the stiffness matrix, the load distribution and the loading function. The index r denotes the dimension of the reduced component. 4.2.10.3.2 Model Reduction of Linear Dynamic Structures Besides our objective to save computational effort by decomposing a structure into its linear and non-linear parts the linear substructures are approximated by dynamically equivalent subsystems. Within the Craig-Bampton method the fixed interface normal modes of each component are defined as a reduced set of modes by restraining all boundary dofs. These modes Φ ik form part of the Craig-Bampton transformation matrix V CB (see section 4.2.10.3.1). In this contribution we will derive the model reduction of linear dynamic systems within a general framework. For simplicity we leave out the superscript s.We start from the ansatz u i = Φ ik q k (4.246) in which u i is the displacement vector, q k is the vector of the reduced system and Φ ik is a rectangular projection matrix of the dimension (i × k). This approach is inserted into the linear equation of motion K ii u i + M ii ¨ u i = b i f(t) . (4.247) This leads to a set of linear equations of reduced dimension (Φ ik ) T K ii Φ ik    K kk q +(Φ ik ) T M ii Φ ik    M kk ¨ q =(Φ ik ) T b i    b k f(t) . (4.248) In the following four different projection-based model reduction methods are summarised. These methods are the Modal Reduction, the Proper Orthogonal Decomposition (POD), a symmetric Pad ´ e-Via-Lanczos algorithm (symm- PVL) and the Load-Dependent Ritz Vectors (LDRV). 4.2.10.3.2.1 Modal Reduction Modal Reduction, also known as Modal Truncation, is the most simple and popular model reduction method. The idea is to solve a subset of the 504 4 Methodological Implementation generalised eigenvalue problem in which Φ ik =[φ 1 , φ 2 , ··· , φ k ] is the reduced modal matrix and Λ kk =diag  ω 2 1 ,ω 2 2 , ···,ω 2 k  is the reduced eigenvalue matrix. After a mass normalisation (Φ ik ) T K ii Φ ik = Λ kk , (Φ ik ) T M ii Φ ik = I kk (4.249) a reduced decoupled differential equation system is obtained: Λ kk q k + I kk ¨ q k = b k f(t) . (4.250) 4.2.10.3.2.2 Proper Orthogonal Decomposition A second possibility is the POD method. This method is also known as empirical eigenvectors, Karhunen-Lo ` eve expansion, principle component analysis, empirical orthogonal eigenvectors, etc. An overview of nomenclatures used in the literature and areas of application are given e.g. in [121]. The mathematical basis for the POD method is the spectral theory of compact, selfadjoint operators which is explained e.g. in the standard text book of [384]. One problem of this ansatz is that even for small systems the eigenvectors of a large spatial covariance matrix have to be calculated. One approach to lower the computational costs is known as the “method of snapshots” ([755]). In this case each POD basis vector φ l = m  j=1 β j w j l =1, ,k (4.251) is generated out of m uncorrelated zero-mean “snapshots” w j . In the latter equation w j = u j − ¯ u describes the deviation of the “snapshot” u j from their temporal mean ¯ u. β j are unknown coefficients which have to be determined. After some derivations and using the assumption that the investigated pro- cess is ergodic (see e.g. [536, 384]) only a reduced eigenvalue problem of di- mension m ×m 1 m W T W β l = λ l β l with W =[w 1 , ··· , w m ] , (4.252) in which W contains the m zero-mean “snapshots” has to be solved. The k basis vectors of the POD φ l = W β l , (4.253) corresponding to the eigenvalues λ 1 >λ 2 > ··· >λ l > ··· >λ k ,resultfrom a linear combination of the zero-mean “snapshots”. 4.2.10.3.2.3 Pad´e-Via-Lanczos Algorithm The Pad ´ e-Via-Lanczos algorithm and the Dual Rational Arnoldi method belong to the Krylov-based model reduction methods. This 4.2 Numerical Methods 505 system-theoretical approach for first order differential equations can also be applied to second order systems. A differential-algebraic equation system K ii u i (t)+M ii ¨ u i (t)=b i f(t) y(t)=c i u i (t) (4.254) is converted by the Laplace transformation to the transfer function H(s)=c i [s 2 M ii + K ii ] −1 b i (4.255) Here the equations are given for a single input single output (SISO) systems. The measurement vector c i of the dimension (1 × i) relates the displacement vector u i (t) to the measured output y(t) of the system. The transfer function (4.255) is re-written and expanded around an expan- sion point σ 2 into a power series (Laurent or Taylor series) H(s)=c i [(K ii − σ 2 M ii )+(σ 2 + s 2 ) M ii ] −1 b i = c i [I ii − (K ii − σ 2 M ii ) −1 (−σ 2 − s 2 )M ii ] −1 (K ii − σ 2 M ii ) −1 b i = ∞  j=0 μ j (−σ 2 − s 2 ) j (4.256) The coefficients μ j = c i ((K ii −σ 2 M ii ) −1 M ii ) j (K ii −σ 2 M ii ) −1 b i are termed “moments”. Thus the method is also called “moment matching” in the literature. An important observation for the Pad ´ e approximation presents the fact that the moments can be computed in a numerically stable fashion by Krylov subspace methods like the Lanczos or the Arnoldi method. For the special case c T i = b i and symmetric, positive definite matrices M ii and K ii [290] show that the reduced systems are stable. According to [486] for mechanical problems purely imaginary expansion points σ = jω c are chosen (ω c is the angular frequency in the centre of the interesting frequency range). Employing a Cholesky decomposition K ii − σ 2 M ii = N ii N T ii and the relations r i = N −1 ii b and G ii = N −1 ii M ii N −T ii the transfer function is transformed into H(s)=r T i [I ii + G ii (σ 2 + s 2 )] −1 r i . (4.257) Using the Lanczos algorithm the matrix G ii is approximated by a tridiago- nalised matrix T kk of the dimension k × k. In the time domain the reduced system results in T kk ¨ q k (t)+[I kk + σ 2 T kk ]q k (t)=r k f(t) y(t)=r T k q k (t) (4.258) 506 4 Methodological Implementation The reduced vector r k =(Φ ik ) T r i is computed according to the projection (4.248). The proposed algorithm for symmetric positive definite system is termed in the following symmPVL. 4.2.10.3.2.4 Load-Dependent Ritz Vectors The method of Load-Dependent Ritz Vectors (LDRV) is an approach of structural dynamics. In the special case that the matrices M ii and K ii are symmetric positive definite matrices, the expansion point is zero (σ = 0), the basis vectors are mass normalised and the input and measuring vectors are identical (c T i = b i ) this algorithm is similar to the SyPVL algorithm proposed by [289]. The LDRV are based on the Lanczos algorithm together with a special start vector. Here the static deflection is used as the first Ritz vector so that all following Ritz vectors may be regarded as the balancing of this initial deflection (see [851]). The advantage of this method is that no eigenvalue problem has to be solved. According to [585] the method delivers the following reduced coupled dif- ferential equation system: T kk ¨ q k + I kk q k = {β 1 , 0, ··· , 0} T f(t) (4.259) Herein the stiffness matrix and the mass matrix are degenerated to an iden- tity matrix I kk and a tridiagonal matrix T kk in generalised coordinates, respectively. If we assume that the load distribution on the structure is con- stant during the simulation, the projected external load vector reduces to b k = {β 1 , 0, ··· , 0} T f(t). The scalar value β 1 =  ϕ T 1 M ϕ 1 is given by the first not mass normalised Ritz vector ϕ 1 . 4.2.10.3.3 Substructuring in the Framework of Nonlinear Dynamics The derivation is based, as displayed in Figure 4.96, on a decomposition of the structure into two arbitrary components. Only with the assembly and the solution of the equation system one of the two components is limited to a reduced linear subsystem. 4.2.10.3.3.1 Discretisation and Linearisation Starting point is the balance of linear momentum of a subsystem s in the reference configuration Div P (s) + ρ 0 b (s) v − ρ 0 ¨ u (s) + F (s) c = 0 s =1, 2 . (4.260) Herein is P (s) the first Piola-Kirchhoff stress tensor, b (s) v the volume force vector, ¨ u (s) the acceleration vector and ρ 0 the mass density in the reference configuration. Additionally interface forces F (s) c have to be introduced. These internal forces only possess non-zero components at the redundant interfaces Γ (s) c . As constraints the equilibrium of the interface forces 4.2 Numerical Methods 507 Component (1) Component (2) Interface no de Internal node Ω (1) 0 Ω (2) 0 Γ (1) c Γ (2) c Fig. 4.96. Decomposition of the structure into two components F (2) c − F (1) c = 0 (4.261) and the compatibility condition of the deformed configuration g = x (2) c − x (1) c = 0 (4.262) has to be fulfilled. x (s) c = u (s) c + X (s) c is the position vector of the deformed configuration at the interface. It is composed of the displacement vector u (s) c and the position vector of the undeformed reference configuration X (s) c . The interface forces may be replaced by the Lagrange-multipliers λ (1) c = λ (2) c = λ c (4.263) since the interface area is of identical size. In the field of contact simulations λ c is named contact pressure. If equation (4.263) is inserted into the strong forms (4.260) and (4.262) in accordance with [473, 856] each subsystem s can be transformed into the weak form g 1 (u (s) , λ)= 2  s=1   Ω (s) 0 P (s) ·Grad δu (s) dΩ (s) 0 +  Ω (s) 0 ρ 0 ¨ u (s) ·δu (s) dΩ (s) 0 −  Ω (s) 0 ρ 0 b (s) v ·δu (s) dΩ (s) 0 −  Γ (s) T ¯ T (s) ·δu (s) dΓ (s) T  +  Γ c λ ·(δu (2) − δu (1) ) dΓ c =0 g 2 (u (s) , λ)=  Γ c (x (2) − x (1) ) ·δλ dΓ c =0 (4.264) . inter- faces have been developed within the scope of reduction methods for large structural dynamic models and the design of single dynamical components of challenging structures. These methods. linear components. 4.2.10.1 Motivation and Overview The importance of simulation tools for the design and lifetime estimation of complex structures increases noticeably. Highly demanding tasks. upper surface of the concrete structure being in front of the supports, there is going to be a structural softening of the concrete block. This interpretation is similiary to the one in [301]