Alijani, Farbod

De Breuker, Roeland

Krüger, Wolf R.

Sinha, Kautuk

Institute of Transport Research:Publications

Kautuk Sinhaⁱ, Farbod Alijani, Wolf R. Krüger, Roeland De BreukerⁱGerman Aerospace Center, Institute of Aeroelasticity, ResearcherⁱDelft University of Technology, Ph.D. CandidateE-Mail: kautuk.sinha@dlr.deA Momentum Subspace-based Model Order Reduction for Finite Element Models in Nonlinear Dynamic AnalysesIntroductionTheoretical formulationAdaptation for dynamicsResults and DiscussionsConclusions and OutlookIntroduction▪ Structural nonlinearities is of interest in several engineering fields.▪ Full finite element (FE) solution can be computationally expensive.▪ Reduced order models (ROM) provide an efficient solution to such problems.▪ The momentum subspace ROM, discussed here, is an adaptation of the Koiter-Newton reduction technique (K. Liang et al., 2013).▪ Extended to dynamics (Sinha et al., 2020) with focus on panel structures.▪ Current studies on cantilevers.Theoretical Formulation▪ The equilibrium equations (statics) are expanded up to the third order in Taylor series.𝑓 𝒖 = ℒ 𝒖 + 𝑄 𝒖, 𝒖 + 𝐶 𝒖, 𝒖, 𝒖 = 𝒇𝐞𝐱𝐭 = 𝐅𝝓 (1)▪ The equilibrium displacement 𝒖 is parametrised by generalized displacements 𝜉.𝐮 𝜉 = 𝐮𝛂𝜉𝛼 + 𝐮𝛂𝛃𝜉𝛼𝜉𝛽 (2)▪ In the reduced subspace, a similar assumption is made for the equilibrium equation.ҧℒ 𝝃 + ത𝑄 𝝃, 𝝃 + ҧ𝐶 𝝃, 𝝃, 𝝃 = 𝜙 (3)▪ Work equivalence to fix the parametrisation.(𝐅𝝓)′ . 𝛿𝐮 = 𝝓′ . 𝛿ξ (4)▪ By regrouping the coefficients of 𝜉, a set of ROM equations are obtained.𝐋 −𝐅−𝐅 𝟎𝐮𝛂ҧ𝐋𝛂=𝟎𝐄𝛂𝐋 −𝐅−𝐅 0𝐮αβഥ𝐐αβ=− Q 𝐮𝛂 , 𝐮𝛃𝟎ഥ𝑪𝛼𝛽𝛾𝛿 = 𝑪 𝒖𝜶, 𝒖𝜷, 𝒖𝜸, 𝒖𝜹 −23𝒖𝒕𝜶𝜷 𝑳 𝒖𝜹𝜸 + 𝒖𝒕𝜷𝜸 𝑳 𝒖𝜹𝜶 + 𝒖𝒕𝜸𝜶 𝑳 𝒖𝜹𝜷▪ The stiffness tensors are obtained as higher order derivatives of strain energy.Theoretical Formulation𝐮 𝜉 = 𝐮𝛂𝜉𝛼 + 𝐮𝛂𝛃𝜉𝛼𝜉𝛽Adaptations for dynamics▪ Full FE equations described by𝑴 ሷ𝐮 + 𝑫 ሶ𝐮 + 𝑳𝐮 + 𝑸𝐮𝐮 + 𝑪𝐮𝐮𝐮 = 𝐅(𝑡)▪ 1st order differential equations in order to perform parametric continuation (AUTO, Doedel, 2007)▪ Hamiltonian formulation to derive the equations of motion.𝐻 𝒖, 𝒑 = 𝑇 𝒖, 𝒑 + 𝑉 𝒖▪ Conservative system - damping and external force excluded initially.▪ An assumption is made for the momentum:𝐩 = 𝐏𝜋, 𝐏 = 𝐌𝚽where P is the basis matrix, 𝜋 is a vector of amplitudes for the momentum vectors.Adaptations for dynamics▪ Potential energy in the reduced subspace:ത𝑉 =12ത𝐿𝛼𝛽𝜉𝛼𝜉𝛽 +13ത𝑄𝛼𝛽𝛾𝜉𝛼𝜉𝛽𝜉𝛾 +14ҧ𝐶𝛼𝛽𝛾𝛿𝜉𝛼𝜉𝛽𝜉𝛾𝜉𝛿▪ Kinetic energy in the reduced subspace:ത𝑇 =12𝜋′ 𝜱′𝐌𝜱 𝜋▪ For a conservative system, ሶ𝝃 =𝜕 ഥ𝐻𝜕𝝅= ഥ𝐌−1𝝅ሶ𝝅 =𝜕 ഥ𝐻𝜕𝝃= − ത𝑳𝝃 + ഥ𝑸𝝃𝝃 + ഥ𝑪𝝃𝝃𝝃ഥ𝐌−1▪ Potential energy in the reduced subspace:ത𝑉 =12ത𝐿𝛼𝛽𝜉𝛼𝜉𝛽 +13ത𝑄𝛼𝛽𝛾𝜉𝛼𝜉𝛽𝜉𝛾 +14ҧ𝐶𝛼𝛽𝛾𝛿𝜉𝛼𝜉𝛽𝜉𝛾𝜉𝛿▪ Kinetic energy in the reduced subspace:ത𝑇 =12𝜋′ 𝜱′𝐌𝜱 𝜋▪ For a non-conservative system, ሶ𝝃 =𝜕 ഥ𝐻𝜕𝝅= ഥ𝐌−1𝝅ሶ𝝅 = −𝜕 ഥ𝐻𝜕𝝃= − ҧ𝐋𝝃 + ഥ𝐐𝝃𝝃 + ҧ𝐂𝝃𝝃𝝃 − ഥ𝐃ഥ𝐌−𝟏𝝅 + ഥ𝝓(𝑡)Adaptations for dynamicsഥ𝐌−1Rayleigh dampingQuadratic damping modelDiscussion of results▪ Test case description1.  Test case 1 : Simply supported square plate (M. Amabili, 2004).2.  Test case 2 : Stiffened plate with free boundary conditions (Sinha et al, 2020).3.  Test case 3 : Ongoing studies, cantilever beam.Discussion of resultsTest case 1: Rectangular plate, simply supported (M. Amabili, 2004)Analysis parameters:l = b = 0.3 mt = 0.001 mDamping ratio 𝜁 = 0.065Applied force 𝑓𝑒𝑥𝑡 = 1.74 N (centre of the plate)Mesh size = 60 x 60Material parameters:E = 70 GPa𝜌 = 2778 Τkg m3Pre-processing: Linear modes analysis Discussion of resultsTest case 1: Rectangular plate, simply supported (M. Amabili, 2004)Comparison to full FE solution in time domain simulation Excitation at ~ 0.997 . 𝜔1Full FE solution time = 274.6  secROM solution time = 3.9 secROM pre-processing = 8.4 secincludes formulation of ROM parameters, stiffness tensors and modal eigenvalue analysisTotal code run-time = 12.3 sec                                                                                            Test case 1: Rectangular plate, simply supported (M. Amabili, 2004)Discussion of results                                                                                                1-DOF modelAnalysis using the software AUTO (Doedel, 2007)Linear modal analysis (pre-processing) = 0.65 secROM formulation time  = 0.99 secAUTO analysis = 5.59 sec  (1560 data points along the solution curve)Difference from reference solution (Amabili, 2004)=  0.43 %Discussion of resultsTest case 1: Rectangular plate, simply supported (M. Amabili, 2004)Convergence study  - increase the number of modes in the reduction subspace.                                                                                                                                                                                                                                   Discussion of resultsTest case 2: Stiffened plate with free boundary conditions (Sinha et al, 2020).Analysis parameters:l = 0.5 m,  b = 0.4 m , t = 0.002 mls = 0.4 m,  bs = 0.008 m , ts = 0.005 m (stiffener)Damping ratio 𝜁 = 0.0012 (initial guess)Applied force 𝑓𝑒𝑥𝑡 = 0.2 – 1 NExcitation frequency = 32 – 40 Hz (sweep)Material parameters:E = 70 GPa𝜌 = 2660 Τkg m31st elastic mode at 36.78 HzExcitation at (x, y) = (0.2, 0.16 m)Discussion of resultsTest case 2: Stiffened plate with free boundary conditions (Sinha et al, 2020).                                                              0.2 N 1.2 N                                                                      Discussion of resultsTest case 3: Ongoing studies, cantilever beam (Pany and Rao, 2002)Analysis parameters:l = 0.693166 mt = 0.001 mDamping ratio 𝜁 = 0.0467Applied force 𝑓𝑒𝑥𝑡 = 0.2 NExcitation frequency = 10.6 rad/s Material parameters:E = 200 GPa𝜌 = 7800 Τkg m3                                                1-cosine profileTest case 3: Ongoing studies, cantilever beam (Pany and Rao, 2002)Discussion of results                                                                        In-plane (IP) displacement                                                                   Out-of-plane (OOP) displacement23.5 % (of length ) maximum OOP deflection  3.2 % (of length ) maximum IP deflection  Simulation time (4-DOF ROM) = 2.4 sec,    simulation time (full Nonlinear) = 223 sec Test case 3: Ongoing studies, cantilever beam (Pany and Rao, 2002)Double the force amplitudeDiscussion of results42.7 % (of length ) maximum OOP deflection  11.5 % (of length ) maximum IP deflection                                                                                                                                    Outlook and Conclusions▪ ROM works well for various boundary conditions.▪ Experiments show us a need of nonlinear damping model.▪ Limited region of ROM validity. ▪ Ongoing studies aim to extend the limit of validity for larger deflections, specially in cantilevers.▪ Intended application towards large scale model reduction of generic FE models.Kautuk SinhaGerman Aerospace CenterEmail: Kautuk.Sinha@dlr.deExtra ▪ Orthogonality conditions derived from constraint equations:▪ Deriving the reduced force from conditions of work equivalence:(𝐅𝝓)′ . 𝛿𝐮 = 𝝓′ . 𝛿ξSubstitute for u: 𝐮 𝜉 = 𝐮𝛂𝜉𝛼 + 𝐮𝛂𝛃𝜉𝛼𝜉𝛽With use of the orthogonality constraints we get,   𝝓 = 𝐟𝐞𝐱𝐭 𝐮𝛂f ′𝛼 𝑢𝛽 = 𝛿𝛼𝛽f ′𝛼 𝑢𝛽𝛾 = 0Extra ▪ Dissipation energy 𝐸𝑑 =12ሶ𝒖′ 𝐃 ሶ𝒖 =12(𝑴−𝟏𝐏𝝅)′ D (𝑴−𝟏𝐏𝝅)𝐸𝑑 = 12ሶ𝝃 ( ഥ𝑴 𝐏′𝑴−𝟏 𝐃𝑴−𝟏𝐏 ഥ𝑴 ሶ) 𝝃▪ von Karman Strain, beam element Discussion of resultsTest case 3: Ongoing studies, cantilever beam (Pany and Rao, 2002)Analysis parameters:l = 0.693166 mt = 0.001 mDamping ratio 𝜁 = 0.0467Applied force 𝑓𝑒𝑥𝑡 = 0.2 NExcitation frequency = 10.6 rad/s Material parameters:E = 200 GPa𝜌 = 7800 Τkg m3                                                                       Pre-processing: Linear modes analysis Test case 3: Ongoing studies, cantilever beam (Pany and Rao, 2002)Convergence analysisDiscussion of resultsNumber of modes𝒖𝟏 [m] 𝒖𝟐 [m] Simulation time [sec]1 -0.0222 0.1627 0.832 -0.0229 0.1643 1.653 -0.0229 0.1645 2.264 -0.0230 0.1645 2.405 -0.0230 0.1646 2.748 -0.0230 0.1646 4.6810 -0.0230 0.1646 6.58

A momentum subspace based model order reduction for finite element models in nonlinear dynamic analyses

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A momentum subspace based model order reduction for finite element models in nonlinear dynamic analyses

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