Presentation on theme: "Coupling Heterogeneous Models with Non-matching Meshes"— Presentation transcript:
1 Coupling Heterogeneous Models with Non-matching Meshes 3/31/2017Coupling Heterogeneous Models with Non-matching MeshesModeling for Matching Meshes with Existing Staggered Methods and Silent BoundariesMike RossCenter for Aerospace StructuresUniversity of Colorado, Boulder24 February 2004
2 Topics of Discussion Introduction of the Research Topic Benefits 3/31/2017Topics of DiscussionIntroduction of the Research TopicBenefitsPlan of Attack for this ResearchProgress (Benchmark Model)Direction for the Future
3 3/31/2017Why am I Here???NSF GrantPIs: Professor Felippa & Professor ParkCrux: Model different physical systems with non-matching meshesUse existing models of individual physical systems.Develop interaction techniques for different modelswith use of a localized connection frames and localized multipliers.
4 Application Drivers MEMS Device (Mechanical Eng.) 3/31/2017Application DriversMEMS Device (Mechanical Eng.)Electrical-Mechanical-Thermal InteractionDam under seismic excitation (Civil Eng.)Fluid-Structure-Soil InteractionReason:NSF granting division is Civil & Mechanical systems (CMS)
5 A Picture is Worth 1,000 Words Multi-physic systemModular SystemsConnected by Localized Interaction Technique (Black Lines)
6 3/31/2017Benefits/Goals:Maintain software modularity through an interface frameProvides great flexibility for different FEM Codes, etc.Custom discretizations and solvers for different physicsSimplify the treatment of non-matching meshesEfficient high fidelity simulationMeaning: resources can be put on critical parts of the problemUser Friendly
7 Plan of Attack Generate a benchmark model 3/31/2017Plan of AttackGenerate a benchmark modelUse current available methodsMatching meshesGenerate a model with localized framesMaintain matching meshesWith nonmatching meshes
8 Current Progress (Benchmark Model) 3/31/2017Current Progress (Benchmark Model)Dam (Brick Elements)Fluid (Spectral Elements)Soil (Brick Elements)Output: Displacements of Dam & Cavitation RegionAssume: Plane Strain (constraints reduce DOF)Only looking at seismic excitation in the x-directionLinear elastic brick elements
10 Structure Equations: x = relative displacements 3/31/2017x = relative displacementsAcceleration (a) is from Berkeley’s PEER databaseDamping modeled with Rayleigh dampingSolved with a Central Difference Method (Explicit). Easy to implement. Also the physics can be represented with small time stepsSilent Boundary on Soil is modeled with a Viscous Damping Boundary MethodSoil and Dam are modeled monolithically with different properties (i.e. Young’s Modulus, etc.)
11 Relative Displacement Concept 3/31/2017Summation of Forces:Matrix Form:
12 Relative Displacement Concept 3/31/2017Relative Displacement ConceptDefine Relative DisplacementsInsert into summation of forces equationMatrix FormKey to remember is that total displacement = earthquake displacement - relative displacement
13 Central Difference Expand Xn+1 & Xn-1 in Taylor Series about time n t 3/31/2017Expand Xn+1 & Xn-1 in Taylor Series about time n tIgnore higher order terms and add and subtract (1) & (2) to yield.Insert these two into the EOMThe Central Difference is said to be second-order accurate. Halving the time step should approximately quarter the error.
14 Stability of Uncoupled Structural Time Integration Central Difference is conditionally stable (General for explicit methods).By going through a Fourier/Spectral Stability:For propagating waves:CFL condition:t must be small enough that information does not propagate across more than one element per time step.
15 JOKE TIMEA football player and an engineer are applying for the same job.The boss said, “Boys, you need to take a test before you can get this job.So they took the test and the next day they came back to see who the boss chose. “Well,” he said, “Both of you got the same score except I’m going to choose the engineer.”The football player complained, “Don’t you think that’s prejudice or something?”“Well,” the boss said,”Let me tell you what happened. Both of your papers were right all the way through until the last question came up, and the engineer answered ‘I don’t know’ and then when I looked at your paper, you answered, ‘Me either’”.
16 Fluid Equations: Assumptions Fluid is inviscid & irrotationalDisplacements are small; thus, density is constantFluid is compressible with a bulk modulusBilinear acoustic fluid (bilinear to account for cavitation)Fluid cannot transmit negative pressuresSystem is initial in static equilibriumSteady body force field (gravity)Goal is to develop continuum fluid models that are discretized with the spectral-element method.
17 Fluid: Momentum Equation Definitions for FluidVolume Force is conservative with Potential EnergyNewton’s Law (F = ma). Momentum Equation
18 Fluid: Momentum Equation Compare to Euler’s EquationFor an acoustic fluid starting at rest one can assume the following in the acceleration term:Thus,
19 Fluid: Displacement Potential Because fluid motion is assumed irrotational, the displacement field can be expressed in terms of the gradient of a scalar function (x,y,z,t).Check Irrotational Condition:
20 Fluid: Equation of Motion Replace acceleration term with this displacement potential in the Momentum Equation.Spatial Integrate:
21 Fluid: Equation of Motion At Static Equilibrium:EOM
22 Fluid: Constitutive Equation For a linear acoustic fluid:K = fluid Bulk ModulusProperty to Characterize CompressibilityIncrease Pressure -> Decrease VolumeK = c2; c = speed of sound of the medium.Define ‘Densified Relative Condensation’Insert: K = c2 &Into Constitutive Eqn.Constitutive Eqn.
23 Fluid: Governing Equation Compare EOM and the Constitutive Equations:We see the following :Apply the wave equation for a Linear Fluid:Get the Governing Equation:
24 Fluid Equations: (Spectral Elements) Dependent field Variables representation within each elementEssences of Spectral Elements is the choice of & quadrature rule by Lagrangian Interpolants with Gauss-Lobatto-Legendre (GLL) quadrature points.Element-node locations are coincident with the quadrature pointsStandard Gauss Quadrature 1-DGLL Quadrature 1-DLinear Shape functions-11-11Locations-1/(3)1/(3)-11
25 Fluid Equations: Discretize governing equation with Galerkin approach Apply Green’s first formulaInsertion of dependent field variable representation, yields the element-level algebraic equations.
26 Fluid Equations: Assemble into global system Q : Capacitance Matrix with Spectral elements (becomes diagonal matrix)H : Reactance Matrixb : boundary-interaction vectorExplicit time integration to solve for s then solve for p
27 Fluid: Bilinear (Cavitation) Cavitation is the spontaneous vaporization of a fluid. It happens when the fluid pressure < vapor pressure.Water’s vapor pressure << atmospheric pressureSimple Mathematical model is that if the total pressure is negative then it is just zero.
28 Explicit Integration for Fluid Add numerical damping to the EOM to reduce frothing = dimensionless damping coefficient (varies from 0 to 1)Modified Solution Advance Process with CavitationInsert into Main Fluid Equation:bn+1 is from predicted structure and NRB displacementsSolve Linear system for sn+1, Remember Q is diagonal
29 Stability of Uncoupled Fluid Time Integration Fourier Stability Analysissee Felippa, Deruntz, sec. 2.4 is the eigenvalue of (H- Q)z=0.Gerschgorin’s theorem can be used to obtain an upper bound on max.
30 Time for a little laughter: “Outside of the killings, Washington has one of the lowest crime rates in the country,” - Washington DC mayor Marion Barry“I’m not going to have some reporters pawing through out papers. We are the President,” - Hillary Clinton“It isn’t the pollution that’s harming the environment. It’s the impurities in our air and water that are doing it.” - Al Gore“It’s no exaggeration to say that the undecided could go one way or another” - George Bush
33 Time Stepping: Staggered Method add subcycling Example: Structure time = t; Fluid time = 2t; Subcycling = 2SSSpbEtc.pn+2upFFtntn+1tn+2
34 Time Stepping: Staggered Method add subcycling Example: Structure time = t; Fluid time = 2t; Subcycling = 2New ustartpressureStructure (seismic displacement, d)Non-Reflecting Boundary (NRB)Fluid Volumedisplacementspn+2Used Subcycling to reduce computational time and have the time increments near the upper bound of the stable time region for the fluid and the structure. Using a Euler Scheme for the prediction.
35 Subcycling Comparison of deflections with subcycling and without. Trying to integrate the different systems with their critical time for stabilityCritical time (fluid,3.5e-3 sec.)>(structure,2.5e-4 sec.)We predict fluid displacements with Euler Scheme.Comparison of deflections with subcycling and without.Dam Crest Movement without subcyclingDam Crest Movement with subcycling
36 Output for Dam Crest Under Seismic Load no Soil With Fluid InteractionWithout Fluid Interaction
37 Output for Dam Crest Under Seismic Load Soil Total Displacement of Dam CrestRelative Displacement of Dam Crest
39 Localized Frame Concept Frames are connected to adjacent partitions by force/flux fieldsMathematically: Lagrange multipliers “gluing” the state variables of the partition models to that of the frame.Lagrange multipliers at the frame are related by interface constraints and obey Newton’s Third Law.
40 Example of Localized Lagrange multipliers (Two-spring system)SubsystemsSubsystem Energy Expressions (Variational Formulation)
41 Total Virtual Work = 0 ( Stationary) Example of Localized Lagrange multipliers(Con’t)The physically represent the interface forcesIdentify Interface ConstraintsTotal Virtual Work = 0 ( Stationary)
42 Example of Localized Lagrange multipliers (Con’t)Equations of MotionComments:Notice the localized multipliers (Lagrange)u1 = uf & u2 = uf (constraint)Last row states that the sum of reaction forces at a node disappear when partitioned nodes are assembled (Newton’s third law)This is just the set up for the transient interaction analysis.
43 Set Sail for the FutureDevelop structure-fluid interaction via localized interfaces with nonmatching meshes.Develop structure-soil interaction via localized interfaces spanning a range of soil media.Develop a localized interface for cavitating fluid and linear fluid.Develop rules for multiplier and connector frame discretization.Implement and asses the effect of dynamic model reduction techniques.
44 Acknowledgments NSF Grant CMS 0219422 Professor Felippa & Professor ParkMike Sprague (Professor Geer’s Ph.D student, now a post-doc in APPM)CAS (Center for Aerospace Structures) CU