Nonlinear Analysis
Nonlinear Analysisfor Improved DesignsNature is nonlinear. Using Marc, capture the inherent nonlinearbehavior of your designs accurately to improve product quality,reduce your testing costs, and improve reliability incorporatingthe true product environment.Marc, the dedicated nonlinear finite element analysis (FEA)solver from MSC Software, is designed to simulate complexnonlinear behavior of engineering materials. Through itsinnovative simulation tools, Marc offers creative solutionsto your toughest nonlinear problems, saves you time, andimproves your productivity.2 MSC Software
Why Marc for Nonlinear FEAAccurate Nonlinear AnalysisProducts are designed to withstand multiple loading scenarios under various environmentalconditions. Hence, it is important to be able to simulate these conditions in order to get a clearerpicture of the product behavior and to improve your designs. Marc offers a broad class of solutionprocedures that you can use to virtually simulate the desired testing conditions and service loads.The industry proven, state of the art methodologies are designed to provide you with accuracy andefficiency that you require in your development process.Nonlinear Material ModelsThe spectrum of engineering materials in use is growing, and so is the need for extensive testingof products designed with these materials. Customizable materials like composites, plastics,elastomers, and new metal like shape memory alloys are helping engineers improve their productswhile operating under engineering constraints.Marc provides users the ability to model a broad range of materials, including metals, shape memoryalloys, super-plastic materials, composites, wood, plastics, rubbers, glass, concrete, powder, metals,and many more. The models incorporate temperature effects, rate effects, and damage so they canaccurately predict the behavior throughout a product’s operating range. Marc also offers an easyapproach to implement new material models that used in state of the art designs and research. So,whether your designs use metals like steel and aluminum or complex materials like composites,solder, or foams, Marc provides you with industry accepted material models to accurately model theirbehavior.Marc Product Brochure 3
Multiphysics and Coupled Analysis for Real World BehaviorAs newer materials are used in designs to benefit from their unique properties, it also becomesimportant to model the physics that control their behavior. For example, piezoelectric materialsaccumulate electric charge when subjected to mechanical stress, and this phenomenon is used insome sensors, actuators, and motors. Also, the phenomenon of Joule heating or resistive heatingdue to the electric energy converted to heat has multiple applications including cooking plates andautomotive defroster grids.The multiphysics capability of Marc, when combined with its superior nonlinear structural analysis,provides more accurate results leading to better and improved designs of your structural systems.Marc can be used to couple structural, thermal, magnetostatics, electrostatics, induction heating,electromagnetic and fluids (laminar flow with small structural deformation) behaviors. The couplingcapabilities are also highly useful to improve the accuracy of manufacturing simulations like welding,curing, and forming.Easy to set up Contact AnalysisAlmost all designs involve interaction with other components within an assembly or interactionbetween assemblies transferring forces across the contact zones. In a nonlinear analysis, forcedirection and magnitude along with the contact zone often change continuously. For analysisaccuracy, it is critical to study the normal contact stresses and the shear stresses transmitted duringthe component interaction.Marc offers a unique approach in tackling this tough problem with the use of smart procedures andeasy set up. Intuitive and easy contact body definitions and automatic contact boundary detectionrelieve users of the burden to create contact interfaces or define master and slave surfaces. SinceMarc is designed to handle both small sliding and large sliding between contact bodies with a singlealgorithm, users do not need to be concerned about the approach they need to use for their specificproblem. Smarter procedures which do not sacrifice performance make for less work for users, whileproviding improved accuracy.4 MSC Software
Failure Analysis for Better ProductsAutomatic Adaptive MeshingPrediction of failure greatly helps with improving your designs, thusextending product life and reducing warranty costs. Because ofthe complex loading a structure could undergo during its life, it isvery expensive to physically test all design for all loading scenarios.Multiple damage models are available for brittle materials such asconcrete, ductile metals, and rubber materials. Modern compositematerials provide new challenges, and Marc provides leading modelsto predict matrix, fiber, and delamination failures. Advanced fracturemechanics capabilities help to predict both crack initiation and crackpropagation. These procedures may be used with a wide range ofmaterials and for virtually all geometries. This provides you the toolsto ensure reliable, safe designs.During certain manufacturing processes or sealing applications, thematerial undergoes severe deformation due to the applied loads orcontact forces. These deformations can be so large that the finiteelement mesh used for analysis could become highly distorted andstall the solver. Marc offers an innovative solution to overcome thisproblem through the use of automatic remeshing.Performance for Improved ProductivityMarc enables higher productivity through solver efficiencies deliveredon multiple fronts. First, Marc helps users achieve higher efficiencythrough state of the art solver technology. With the falling cost ofmulti-core systems, parallel processing is increasingly accessible tosmall and medium businesses operating in a desktop environment.To help users make full use of their hardware, Marc provides parallelsolvers to everyone at no additional cost. Finally, through its uniqueimplementation of domain decomposition method whereby the modelis broken into smaller pieces and solved in entirety on separateprocessors, Marc goes beyond traditional parallelization techniquescommon in industry. With this approach, all the steps of FEA processincluding, input, stiffness matrix assembly, matrix solution, stressrecovery, and output are executed in parallel on shared memory ordistributed memory systems helping achieve superlinear scalabilityfor your large models.During the analysis, if elements become severely distorted, Marcautomatically creates a new mesh from the deformed boundary.The state of the material (stresses, strains, deformations) along withcontact conditions are transferred to the new, well structured mesh,and the analysis is carried further with the new mesh. This processis repeated as many times as necessary without user interventionallowing you to solve tougher, complex nonlinear problems.CustomizationMarc provides the ability to customize the analysis software throughthe use of user subroutines. Over 175 are available to incorporateadvanced material models, load and boundary conditions, or elementtechnology. The GUI – can be easily customized using Python.Marc Product Brochure 5
Easy to Use, Integrated User InterfaceTightly integrated user interface to help you build, analyze, and postprocessthe FEA models with ease. Designed with focus on supporting nonlinear analysis, youbenefit from time-saving tools and processes implemented in the Marc user interface.Geometry and MeshingMaterial Modeling“Benchmarkinvestigations showedclearly that Marc wasthe best solution inevery aspect” Import CAD files in neutral formats ofACIS, STEP, STL, IGES, VDAFS, andParasolid Define nonlinear material parameters’Dr. Martin English,Hadley Group Technology Create 1D, 2D, and 3D lower andhigher elements using automatic meshgenerators Import geometry in native CAD formatsof CATIA V4, CATIA V5, Pro/E Wildfire,Solidworks, Unigraphics, Inventor, andDXF Create and edit curves, surfaces, andsolids Refine and edit mesh entities Curve fit experimental data to obtainparameters for elastomers Specify temperature dependent materialproperties Define properties as functions of otherindependent variablesLoads and Boundary Conditions Define required forces, moments,displacements, pressures, and rotations Create springs and dashpots for analysis Specify boundary conditions relevant toeach multiphysics model set up Create discrete connecters such asRBE2, RBE3, Servo-links, CBUSH,CFAST, and CWELD Apply boundary conditions to geometricentities to easily transfer to associatedmesh Insert nodes or elements within otherelements Define boundary conditions as functionsof other independent variables Group loads and boundary conditionsinto load cases easily6 MSC Software
ContactModeling Tools Easily define deformable and rigid contactbodies Easily create bolt models and loads forassembly analysis Automatically create contact bodies fromCAD assembly models Create gas filled closed cavities andanalyze their influence on structuralresponse Customize contact interaction usingcontact tables Define friction coefficients and otherparameters for contact analysis Supports large deformation interference fit Support contact in multiphysicsapplications.Post-processing Obtain plots of results in format you desireincluding contour plots, contour lines,cutting planes, iso-surfaces, and more Plot time variation of any of the resultssaved at any arbitrary location Transfer pre-state from one analysis toanother Map results from axisymmetric analysis toa 3D model Create symmetry and cyclic symmetryboundary conditions Define welding paths and filler elements forwelding analysis Activate and Deactivate elements Define crack tips for fracture analysis Define electrical coil windings forelectromagnetic analyses Use path plot to review the spatial variationof any result along any arbitrary path Create images and movies for reports andpresentations Track material particle flowMarc Product Brochure 7
Nonlinear Structural AnalysisReduce your physical testing costs with reliable, robust simulation of structuralresponse under a wide spectrum of loading scenarios and environmental conditions.“Marc gives usconfidence in ourresults and offers acomplete list offeatures. I am verysatisfied with theresults of our analyses”Luigi Piccamiglio, Invernizzi PresseStatic AnalysisDynamic Analysis Perform linear and nonlinear staticanalysis to virtually test your designs Perform natural mode analysis ofstructures to determine structural stabilityunder dynamic loads Include advanced nonlinear materialmodels Incorporate both large deformation andlarge strain behavior Accurately model nonlinear boundaryconditions including follower forceeffects, foundations, and contact Perform creep simulations to determinethe long term response of the structure. Perform post-buckling analysis toperform stability studies Determine the inertia relief force tobalance free structures Perform steady-state rolling analysis oftires Perform mechanical wear analysis due tofriction Export or Import DMIG files forcompatibility with MSC Nastran Perform global-local analysis to bettercapture local behavior8 MSC Software Conduct frequency response analysissubjected to harmonic loads orrandom vibrations to analyze structuralperformance Include advanced damping models thatincorporate frequency and deformationdependent damping observed in rubberand plastics Obtain insight into dynamic performanceof structures through transient analysis Gain improved accuracy through accuratemodeling of contact, nonlinear materials,and loading conditions Create Modal Neutral Files (MNF) thatmay be shared with Adams includingnonlinear preload
Multiphysics and Coupled AnalysisThe multiphysics capability of Marc, when used with its superior nonlinear structuralanalysis, provides more accurate results leading to better and improved designs ofyour structural systems.“Marc is a multiphysics tool withcapabilities for flow,heat, mechanical stress,and electrical modeling.In addition, the openstructure of Marcallows customization- databases, userroutines, and so on”Dr. Moe Khaleel,Pacific Northwest National LaboratoryHeat TransferThermomechanical Coupling Perform steady-state and transientanalysis for one-, two-, andthreedimensional bodies Analyze structural response due totemperature changes in the environmentand thermal gradients in the structure Obtain temperature distributions in astructure for linear and nonlinear heattransfer problems Model heat generation due toplasticity and friction between differentcomponents for accurate physics Model nonlinearities includingtemperature-dependent properties,latent heat (phase change) effect,heat convection in the flow direction,and nonlinear boundary conditions(convection and radiation) Incorporate heat generation due to curingin composite manufacturing. Compute radiation view factors faster andmore accurately Simulate thermal degradation of ThermalProtection Systems (TPS) with advancedpyrolysis model Perform ablation analysis for spacesystems, brakes, and bio-medicalapplications Compute heat fluxes across multiplecomponents that come into contact Simulate the influence of annealing Simulate the effects of changes tothermal boundary conditions due to largedeformationsAcoustics and CoupledAcoustic-Structural Analysis Perform acoustic analysis in rigid anddeformable cavities Calculate fundamental frequencies of thecavity, as well as the pressure distributionin the cavity Compute the effect of the acousticmedium on the dynamic response ofthe structure and the influence of thestructure on the dynamic response of theacoustic medium Use contact body definition to simplifymodel creation using dissimilar meshesMarc Product Brochure 9
Fluid MechanicsElectromagnetic AnalysisPiezoelectric Analysis Perform laminar flow analysis withNavier-Stokes equations Perform transient and harmonic fullycoupled electromagnetic analysisto calculate electrical and magneticfields subjected to external excitation Simulate piezoelectric effect ofcoupling of stress and electric field ina material Conduct studies that involvefluidthermal coupling, fluid-solidcoupling and fluid-thermal-solidcoupling Solve steady state and transient flowproblems in two- and three-dimensionElectrostatics andMagnetostatics Evaluate electric fields and magneticfields in a body or medium Compute electric potential field,electric displacement vectors,magnetic induction, magnetic fieldvector, and more to gain insight Model infinite domain with semiinfiniteelements for improved accuracy Determine the capacitance betweenelectrical conductors in electrostaticanalyses Calculate the inductance due to wiresor coils in magnetostatic analyses10 MSC Software Compute magnetic permeabilityas a function of magnetic field in atransient analysis Calculate magnetic flux density,magnetic field vector, electric fluxdensity, and electrical field intensityalong with potential, nodal charge,and current Solve for nodal displacements andelectrical potential simultaneously Perform static, transient dynamic,harmonic, and eigenvalue analysis tobetter understand material response Couple with heat transferanalysis to perform a coupledthermalpiezoelectric analysis
Coupled Electrostatic-StructuralAnalysisCoupled Electrical-ThermalMechanical AnalysisCoupled Electromagnetic-ThermalAnalysis (Induction Heating) Simulate the influence of Coulomb forceson structural components and deformationinfluence on electrostatic field Simulate structural response due to Jouleheating effects Simulate induction heating with staggeredapproach of harmonic electromagneticanalysis followed by thermal analysis Model contact between different bodiesand simulate influence of their interactionon the fieldCoupled Thermal-Electrical Analysis(Joule Heating) Compute heat generated due to electricflow in a conductor Model temperature dependent resistanceand internal heat generation as a functionof the electrical flow Determine the resistance of the device Account for nonlinearities that arisedue to convection, radiation, andtemperaturedependent thermalconductivity and specific heat Simulate structures with nonlinearities dueto geometric and material behavior Compute induced current which generateheat and heat flux Incorporate temperature dependency formaterial data for improved accuracy Use contact analysis to analyze interactionbetween multiple componentsCoupled ElectromagneticStructural -Thermal AnalysisCoupled MagnetostaticStructural Analysis Simulate induction heating with staggeredapproach of harmonic electromagneticanalysis followed by thermal-stressanalysis Simulate the influence of Lorentz forces onstructural components and deformationinfluence on magnetostatic field Model contact between different bodiesand simulate influence of their interactionon the field Compute induced current which generateheat and heat flux Incorporate temperature dependency formaterial data for improved accuracy Accurately determine the surface strainsand stresses Utilize either adaptive meshing or adualmesh approach to model large motionthrough the air Predict the amount of power required forsurface treatmentMarc Product Brochure 11
Material Models for Accurate ResultsSimulate materials accurately with the comprehensive library of engineeringmaterial models that range from simple linear elastic materials to complex timeand temperature-dependent materials, providing you with accurate results.Elastic-Plastic Materials“The simulationperformed using Marchas given us a goodfeeling for the behaviorof the Ramspol rubberdam and has improvedour knowledge ofthis kind of flexiblestructures” Model isotropic, orthotropic, andanisotropic elastic materialsHans Dries, HollandscheBeton-en Waterbouw Incorporate thermal and rate effects withthe Power law models, Johsnson-Cook,Cowper-Symmonds, or Kumar models Use isotropic, kinematic, or combinedhardening rules to model work hardening Model plasticity accurately selectingfrom a choice of yield criteria, includingvon Mises, Hill anisotropic yield function,Barlat’s, Mohr-Coulomb, and ExponentialCap Model Simulate granular material behavioraccurately through powder materialmodel Study the effects of cyclic plasticity onplastic strains using the Chaboche model Model the damage of ductile metalsTime-dependent InelasticMaterial Models Analyze creep which is an importantphenomenon at higher temperature stressanalysis Study the combined effects of plasticityand creep on structural behavior Simulate the dilatation creep (swelling) ona structure for improved accuracy12 MSC Software Analyze isotropic and anisotropicbehavior through thermorheologicallysimple viscoelastic material model Incorporate frequency dependentdamping models based upon theviscoelastic nature of rubber and plastics Improve result accuracy with materialmodels of choice to model viscoplasticbehavior Include temperature effects in creep andrelaxation studies for improved resultsComposite Materials Model shell structures with layerscomposed of different materials or layersof the same anisotropic material withvarious layer thicknesses and differentorientations Incorporate both linear and nonlinearmaterial properties in each layer Model the degradation of the materialusing progressive failure analysis (PFA) Model the delamination of materials Use any of the multiple orientationspecification options that providemodeling flexibility
Nonlinear Hypoelastic MaterialElastomersSpecial Class Materials Model a generalized nonlinear elasticmaterial to simulate behavior of materialswith nonlinear stress-strain relationship Choose from multiple material modelsincluding Generalized Mooney-Rivlin,Ogden, Arruda-Boyce, Bergrstrom-Boyce,Gent, and Marlow to represent elastomers Model complex, multi-layer gasketelements that are often made of differentmaterials of varying thickness Use any of the multiple modeling options,which include stress-strain relationshipbased model, strain invariant model,principal strain space model, bimoduluselasticity which could incorporate notension, limited tension, no compression,or limited compressionShape Memory Material Simulate the reversible, thermoelastictransformation of shape memory alloysbetween high temperature austenitic phaseand low temperature martensitic phase Model transformation induced deformationand irreversible permanent deformation ofthe martensite Easily obtain required material parametersfor elastomeric material models throughintegrated curve fitting functionality Analyze thermal effects and creep/relaxation of elastomers through largestrain viscoelastic model Model complex loading and unloadingbehavior Perform mechanical, thermal, orthermomechanically coupled analyses fordesired accuracy and required outputs in agasket analysis Analyze low tension materials like concreteto accurately determine structural strengthand behavior under complex loads Incorporate rebars to model reinforcedstructures like composites, biologicalmaterials, or reinforced concrete Analyze soil material behavior usingformulations like Drucker-Prager, MohrCoulomb, Cam-Clay, Exponential Cap, andtheir variations Choose from mechanical andthermomechanical shape memory modelsdepending on your loading conditionsMarc Product Brochure 13
Easy to use and Powerful Contact AnalysisUtilize Marc’s unique contact modeling approach and technology to solveyour tough analysis problems that could include large deformation and rotation.Take the hard work out of contact model creation with easy, intuitive set up.Contact Setup“Marc is a very goodproduct to study bothcontact and the thermomechanical behaviorof contacting parts”Luigi Piccamiglio, Invernizzi Presse Define contact bodies (linear and quadratic)by selecting elements without the need todefine contact boundary elements Reduce contact set up time throughautomatic contact boundary detection Reduce your modeling time and improveaccuracy by using the CAD geometrysurfaces and curves to define rigid bodies Apply velocities, displacement, loads,moments, and rotations to rigid bodies Use rigid bodies to apply symmetryboundary conditions Use contact tables to customize contactdetection between bodies Accurately analyze contact between shellsand shells or shells and solids Predict contact on both sides of shells Analyze beam contact with arbitrary beamcross-sections Analyze tube-in-tube contact behaviorobserved in the oil, automotive, andbiomedical industries using beam elementsbut with 3D contact behavior Use segment-to-segment contact methodfor smoother results contours and toovercome the limitations of master-slavecontact method Easily model interference fit and overclosure14 MSC Software
FrictionContact in Coupled AnalysisContact Results Specify different friction coefficientsbetween different contact pairs Perform thermal contact analysis withoutstructural analysis to study heat transferbetween different bodies Analyze the progression of contactbetween contact bodies Choose from multiple friction models tosuit your requirements – Coulomb friction,shear friction, stick-slip friction, andbilinear friction model Use glue option to analyze contact withvery high friction by preventing relativetangential motion Study effects of friction generated heaton structural behavior with support forcoupled analysis Analyze flow of current between contactbodies for accurate result Study the contact forces and pressuresand map the distribution in the contactregions Sum up the contact forces on a body toevaluate the total forces Analyze glue failure between contactbodies by automatic glue deactivationMarc Product Brochure 15
Automatic Adaptive Meshing for Improved ConvergenceSave time and effort by taking advantage of automatic mesh updates in Marc.Get a quality mesh when it is needed most.“(Using Marc) we wereable to substitutenumerical simulationfor the physicaldevelopment runs,which was very costeffective”Charles Courlander,Gaurdian Automotive LuxembourgLocal AdaptivityGlobal Adaptivity Increase the number of elements andnodes to improve accuracy of thesolution Automatically replace the distorted meshof a contact body with a better qualitymesh Automatically update the mesh bysubdividing elements that satisfy any ofseveral criteria available Map the results of the old mesh to thenew mesh without user intervention Update boundary conditions on newlycreated elements and faces Attach the nodes to a curve or surfaceassociated with rest of the mesh tosmoothen the boundary Manually update the mesh at a desiredpoint of solution and map the results tothe new mesh Automatic transfer of boundaryconditions to the new mesh Solve 2D and 3D models with improvedconvergence and accuracy Generate new refined meshes basedupon the solution quantities, such asplastic strain or strain energy density. Utilize adaptive meshing to capture crackpropagation16 MSC Software
Manufacturing ProcessesSimulate manufacturing of your product to optimize your manufacturing processand to incorporate the effects of residual stresses in product analysis.“We were able tocompare a virtual videoof the deformationgenerated. with realvideo footage shot inone of our plants.The match wasimpressive”Nicholas Behr, Setforge EngineeringFormingComposites Simulate forming operations like brakeforming, hydroforming, glass forming,stretch forming, and roll forming Analyze progression of curing andoptimize the design and processparameters Easy model multi-stage processes Predict shrinkage caused by curing ofresin Compute and minimize residual stressesfor improved designs Accurately calculate the spring-backduring the manufacturing process Perform cure-thermal-mechanicalcoupled analysis to predict residual stressbuildup during the curing process Analyze rubber extrusion and theoptimize the process Use automatic remeshing to obtainaccurate results and improvedconvergenceOther Processes Simulate welding process and its effectson final structural deformation Import NC machine file and analyzethe deformations induced by materialremoval Study assembly process with boltinstallation and riveting Remove the residual stresses withannealing for subsequent product testingMarc Product Brochure 17
Wear and Failure AnalysisPredict product failures to improve your designs and reduce warranty costs.Reduce development costs by limiting costly destructive testing.“The capacity in Marc.to automaticallyinternally restart theincrement after crackgrowth, and hencecapture multiple crackgrowth events in oneincrement, was highlybeneficial in termsof e ModelsFracture Mechanics Compute damage accumulation in ductilematerials using the Bonora or Gursovoidgrowth-based model Evaluate energy release rate using Lorenzimethod or Virtual Crack Closure Technique(VCCT) Calculate damage factors using Lemaitre,Cockroft-Latham, or Oyane model Evaluate J-integral based on domainintegration method and compute stressintensity factors for the three modes Simulate Mullin’s effect and damageaccumulation due to polymer chainbreakage in elastomers Combine rubber damage model withviscoelastic behavior using the BergstromBoyce model Study brittle cracking and crushing inconcreteComposite Failure Analyze laminate failure by choosing fromindustry accepted failure criteria thatinclude maximum stress and strain criteria,Hill, Hoffman, Tsai-Wu, Hashin, HashinFabric, Hashin tape, Puck, and StrainInvariant Failure Theory (SIFT) failure criteria Analyze progressive failure of composites Study delamination failure using CohesiveZone Modeling of the bonding betweenlamina Compute energy release rates for brittle andductile materials at small and large strainsAutomatic Crack Propagation Analyze crack growth under direct loadingor fatigue loading Simulate crack growth along edges oralong contact region Simulate crack propagation either throughthe elements or utilizing automatic adaptivemeshingWear Analyze mechanical erosion due to impactof particles, part vibration, or shear stressdue to flow Study erosion due to pyrolysis involvingdecomposition of material due to thermalprocess Model thermochemical ablation of materialsubjected to high thermal fluxes Study tool effectiveness by simulating toolwear18 MSC Software
Efficient Solvers and Parallel ProcessingFor Higher ProductivityTake advantage of efficient matrix solvers and unique parallel processingmethodologies to get results faster.Giuseppe Simonetti, BogWarner Inc.Parallelization Use solvers designed for bestperformance, including multifrontal,Pardiso, MUMPS, and CASI iterativesolver Speed up your solution by using allavailable cores on multicore desktops Use iterative solvers to run larger modelswithout high memory usage Use out-of-core option to make efficientuse of disk space to solve large models Benefit from NVIDIA GPU for improvedperformance Solver specific bandwidth optimization toreduce storage space Obtain true parallelization throughDomain Decomposition Method byparallelizing all phases of the solution namely input, assembly, solution, stressrecovery, and output Automatic and manual decomposition ofthe model for parallelization Make the best use of your hardware whether a shared memory parallel ordistributed memory parallel systemScalability“I am very impressedwith the capabilities.I’m very satisfied withthe Pardiso Solver andthe speed increase itprovides”Solvers0246810DomainsMarc Product Brochure 19
Marc Advanced Nonlinear and Multiphysics AnalysisMSC Software is one of the t
Marc, the dedicated nonlinear finite element analysis (FEA) solver from MSC Software, is designed to simulate complex nonlinear behavior of engineering materials. Through its innovative simulation tools, Marc offers creative solutions to your toughest nonlinear problems, saves you time, and improves your productivity.
Nonlinear Finite Element Analysis Procedures Nam-Ho Kim Goals What is a nonlinear problem? How is a nonlinear problem different from a linear one? What types of nonlinearity exist? How to understand stresses and strains How to formulate nonlinear problems How to solve nonlinear problems
Third-order nonlinear effectThird-order nonlinear effect In media possessing centrosymmetry, the second-order nonlinear term is absent since the polarization must reverse exactly when the electric field is reversed. The dominant nonlinearity is then of third order, 3 PE 303 εχ The third-order nonlinear material is called a Kerr medium. P 3 E
Outline Nonlinear Control ProblemsSpecify the Desired Behavior Some Issues in Nonlinear ControlAvailable Methods for Nonlinear Control I For linear systems I When is stabilized by FB, the origin of closed loop system is g.a.s I For nonlinear systems I When is stabilized via linearization the origin of closed loop system isa.s I If RoA is unknown, FB provideslocal stabilization
oriented nonlinear analysis procedures” based on the so-called “pushover analysis”. All pushover analysis procedures can be considered as approximate extensions of the response spectrum method to the nonlinear response analysis with varying degrees of sophistication. For example, “Nonlinear Static Procedure—NSP” (ATC, 1996; FEMA, 2000) may be looked upon as a “single-mode .
Nonlinear analysis for improved designs Nature is nonlinear. Using Marc, accurately capture the inherent nonlinear behavior of your designs to improve product quality, reduce your testing costs, and improve reliability incorporating the true . Perform global-local analysis to better capture local behavior Dynamic analysis
eigenvalue buckling analysis, nonlinear stress analysis, and graphical post-processing. In this paper a brief description of CALEB version 1.4 and of its main features is presented. INTRODUCTION CALEB is a nonlinear finite element program for geometric and material nonlinear analysis of offshore platforms and general framed structures.
in the general nonlinear case via interval analysis. Key Worda--Bounded errors; global analysis; guaranteed estimates; identification; interval analysis; nonlinear equations; nonlinear estimation; parameter estimation; set theory; set inversion.
Tutorial on nonlinear optics 33 rank 2, χ(2) a tensor of rank 3 and so on. P 1(t) is called the linear polarization while P 2(t)andP 3(t) are called the second- and third-order nonlinear polarizations respec- tively. Thus, the polarization is composed of linear and nonlinear components. A time varying nonlinear polarization
