Faculty of Civil, Geo and Environmental Engineering Chair for Computation in Engineering Prof. Dr. rer. nat. Ernst Rank Dynamic Substructuring using the Finite Cell Method Sebastian Johannes Buckel Bachelor's thesis for the Bachelor of Science program Engineering Science Author: Sebastian Johannes Buckel Supervisor: Prof. Dr.rer.nat. Ernst Rank Dipl.-Ing. Tino Bog Nils Zander, M.Sc. Date of issue: 01. June 2014 Date of submission: 07. November 2014 Involved Organisations Chair for Computation in Engineering Faculty of Civil, Geo and Environmental Engineering Technische Universität München Arcisstraÿe 21 D-80333 München Declaration With this statement I declare, that I have independently completed this Bachelor's thesis. The thoughts taken directly or indirectly from external sources are properly marked as such. This thesis was not previously submitted to another academic institution and has also not yet been published. München, November 2, 2014 Sebastian Johannes Buckel Sebastian Johannes Buckel Am Schuÿ 54 D-83646 Bad Tölz e-Mail: [email protected] V Contents 1 Introduction1 1.1 Motivation......................................1 1.2 Scope of the Work.................................2 2 The Finite Cell Method for Structural Dynamics3 2.1 Weak Formulation of the Linear Elastodynamic Equation...........3 2.2 High Order Shape Functions............................6 2.3 The Embedded Domain Concept.........................6 2.4 Adaptive Integration................................8 2.5 Weakly Enforced Boundary and Interface Constraints..............9 2.6 Newmark Method.................................. 10 3 Generation of Superelements using the Finite Cell Method 13 3.1 Craig-Bampton Method.............................. 13 3.1.1 Component Modes............................. 14 3.1.2 Craig-Bampton Transformation...................... 14 3.1.3 Substructuring and Super Elements.................... 16 3.2 Implementation................................... 16 4 Numerical Examples 19 4.1 Periodic Excitation of a 2D Bar.......................... 19 4.2 Impulsive Excitation of a 2D Bar......................... 29 4.3 Finite Cell Method meets Craig-Bampton Method............... 36 5 Summary and Conclusions 41 1 Chapter 1 Introduction 1.1 Motivation Static and dynamic computations in civil, automotive and aerospace engineering, the simu- lation of enviromental ows and aerodynamics, or researches in the eld of heat transfer and nano engineering, they all often leads to a partial dierential equation (PDE). In general, this equation can not be solved analytically. Hence, methods had been introduced in order to solve the equation numerically. One of the most important numerical methods for science and technology is the Finite Element Method (FEM) [14]. Today, there are more than 1000 FEM commercial software systems, which generate an annual revenue of over 1 Billion US$. The FEM was developed in the 1950s and because of the increasing power of computers in the late 1970s the method became attractive for the calculation of technical problems [2]. One of the limiting factors of the FEM is the runtime. Although the power and computation rate of the computers was multiplied over the years, there was always the aim to obtain simplications. Thus, more complex problems should be calculated in less time. Especially in computational dynamics the combination of large, detailed models and the dis- cretization in time leads in a very long calculation time. For each time step a modied system of equations has to be solved. Hence, for dynamic problems a simplication is very useful. One approach in computational mechanics is the use of structural elements, such as beams of shells. The combination of various structural elements should simulate the behaviours of the original problem. Especially for voluminous geometries this method isn't suitable. An- other idea is the reduction of the number of degrees of freedom (DOF). Therefore, the DOFs are divided in master and slave DOFs. After the reduction the master DOFs are still avail- able. Hence, the master DOFs have to be chosen carefully [20]. For static problems the Guyan-Reduction [9] is suitable. This method also holds good for dynamic problems, if the periodic loading is low frequency or the loading isn't applied suddenly [2]. But for problems in the high frequency range another method has been developed, the Craig-Bampton Method [1]. Additionally to the physical master DOFs also modal DOFs are considered for a better dynamic behaviour [4]. For very complex models the concept of dynamic substructuring is introduced. The model can be divided in dierent substructures. Then each substructure can be improved and reduced independently. Afterwards the substructures are connected to the original model. A reduced substructure segment is called super element. 2 1. Introduction A further very limiting factor in the FEM is the mesh generation. Especially if high quality criteria are required to the mesh, the prepossessing becomes very time-consuming. In order to overcome this problem the Chair for Computation in Engineering of the Technische Universität München developed the Finite Cell Method (FCM) [5, 12, 15, 17]. Because of the special domain approach many quality criteria are fullled implicitly. This thesis combines the FCM and the Craig-Bampton Method and constitutes a powerful way to handle with complex problems. 1.2 Scope of the Work The combination of the FCM and Guyan-Reduction for static problems is already analysed by Ruchti [16]. But the Craig-Bampton Method using the FCM for dynamic problems is presented in this thesis. In chapter2 the Finite Cell Method is introduced. In the context of the work two dimensional, linear elastic problems are considered. A special focus is put on applying weakly enforced constraints. Therefore, the penalty method is introduced. The time integration is realized by the use of the Newmark Method. In chapter3 the Craig-Bampton Method is presented. This chapter is divided in two parts. At rst the theory is described. Then the way is illustrated, how the method is integrated in the FCMLab MATLAB Toolbox [22]. Finally various simulations, which were calculated in the context of the thesis, are visualised and analysed. The conclusive simulation joins the Finite Cell Method and the Craig-Bampton Method. 3 Chapter 2 The Finite Cell Method for Structural Dynamics A very common numerical method in order to solve problems in structural mechanics is the Finite Element Method (FEM) [2, 10, 18]. One of the limiting factors of the FEM is the mesh generation. Especially for non-trivial geometries in combination with high quality mesh criteria the mesh generation may becomes very complex and time-consuming. In order to solve this problem the Finite Cell Method (FCM) was developed by the Chair for Computation in Engineering (CiE) [5, 12, 15, 17]. The concept of the FCM is presented in this chapter. Based on the governing equations of linear elastodynamics, introduced in section 2.1, the concepts of high order shape functions, section 2.2, and the embedded domain approach, section 2.3 are described. Because of the special domain approach a adaptive integration scheme is introduced in section 2.4. Furthermore, weakly enforced boundary and interface constraints are presented in section 2.5. The resultant time-dependent equation can be solved by the Newmark method, which is introduced in section 2.6. 2.1 Weak Formulation of the Linear Elastodynamic Equation The standard problem in structural dynamics is a elastic continuum with the domain Ω and its boundary δΩ. There are two dierent types of boundaries, Dirichlet ΓD and Neumann ΓN boundaries, which have to satisfy the following conditions: ΓD [ΓN = δΩ and ΓD \ΓN = f0g. According to Wall [19] the following initial-boundary value problem (IBVP) describes the properties for the linear elastodynamic problem. The formulation of the IBVP is called strong formulation. Field equations: Balance equation (BE): r · σ + b = ρu¨ in Ω × [t0; tend] (2.1a) Constitutive equation (CE): σ = C : " (2.1b) 4 2. The Finite Cell Method for Structural Dynamics Kinematic equation (KE): 1 T (2.1c) " = 2 (ru + (ru) ) Velocities: d (2.1d) u_ = dt u Accelerations: d d2 (2.1e) u¨ = dt u_ = dt2 u where σ is the stress tensor, " the strain tensor, C the forth order elasticity tensor, b the body force vector, ρ the density and u the displacement vector. Boundary conditions (BC): Dirichlet BC (DBC): u = u^ on ΓD × [t0; tend] (2.1f) Neumann BC (NBC): σ · n = t^ on ΓN × [t0; tend] (2.1g) where u^ is the prescribed boundary displacement vector, n the unit normal vector and t^ the given boundary traction vector. Initial conditions (IC): u(x; t0) = u^0 in Ω (2.1h) u_ (x; t0) = u^_ 0 in Ω (2.1i) where u^0 is the given initial displacement and u^_ 0 the given initial velocity. In real life problems, it's usually not possible to nd an analytical solution of the strong formulation of the IBVP. Hence, approximation methods are needed to obtain a solution. A very suitable method for structural problems is the nite element method, which is based on the weak formulation of the IBVP. Therefore, the balance equation (2.1a) is integrated over the domain Ω and weighted by a test function v 2 V [10, 19]: Z (r · σ + b^ − ρu¨) · v dΩ v 2 V; (2.2) Ω 1 with V = fv j v 2 H ; v(xD) = 0 : xD 2 ΩDg; (2.3) 1 U = fu j u 2 H ; u(xD) =u ^ : xD 2 ΩDg (2.4) and H1 are the functions of the rst order Sobolev space. This equation is also called equation of virtual work and its trail function u 2 U, which satises the equation for any test function v 2 V, is called weak solution. When using the innite-dimensional Sobolev space, the weak solution is exact. For further considerations the divergence theorem and the approach for the test function at the Dirichlet boundary is applied on the equation of virtual work (2.2).