Nonlinearities are common in most real-world problems and represent some of the most challenging aspects of engineering analysis. To help users model and solve these problems, ANSYS has a wide range of features and capabilities for handling the most common types of nonlinearity. Changing status or contact nonlinearity — Many common structural features exhibit nonlinear behavior that is status-dependent. Status changes might be directly related to load, or they might be determined by some external cause. Situations in which contact occurs are common to many different nonlinear applications. Contact behavior, such as separation and sliding with frictional effects, introduces nonlinearity into the analysis. Geometric nonlinearity — If a structure experiences large deformations, its changing geometric configuration can cause the structure to respond nonlinearly. Geometric nonlinearity is characterized by “large” displacements and/or rotations. Small deflection and small strain analysis assume that displacements are small enough that the resulting stiffness changes are insignificant. In contrast, large strain analysis account for the stiffness changes that result from changes in an element’s shape and orientation. The large strain feature is available in most of the solid elements (including all of the large strain elements) as well as in most of the shell and beam elements. ANSYS also handles two other types of geometric nonlinearities: stress stiffening and spin softening. For thin, highly stressed structures, such as cables and membranes, the out-of-plane stiffness of a structure can be affected significantly by the state of in-plane stress in that structure. Stress stiffness is the coupling between in-plane stress and transverse stiffness. Spin softening softens the stiffness matrix of a rotating body for dynamic mass effects. The adjust- ment approximates the effects of geometry changes due to large deflection circumferential motion in a small deflection analysis. Spin softening is used in conjunction with prestressing, which is caused by centrifugal force in the rotating body. Material nonlinearity — Nonlinear stress–strain relationships are a common cause of nonlinear structural behavior. Many factors can influence a material’s stress–strain properties, including load history (as in elastoplastic response), environmental conditions (such as temperature) and the amount of time that a load is applied (as in creep response). ANSYS handles numerous material-related factors that cause a structure’s stiffness to change during the course of an analysis ranging from anisotropic behavior, nonlinear stress–strain relationships, dependency on time, rate of strain and certain coupled physics effects such as piezoelectric and Seebeck effects, to name a few. By Achuth Rao, Ph.D. Product Manager ANSYS, Inc. Nonlinear history tracking option monitors results in real time during solution. ANSYS Nonlinear Technology Powerful new capabilities are aimed at studying complex nonlinear behavior in mechanical systems. 18 www.ansys.com ANSYS Solutions | Volume 7, Issue 3 2006
Transcript
Nonlinearities are common in most real-world problems and represent some of the most challengingaspects of engineering analysis. To help users modeland solve these problems, ANSYS has a wide range of features and capabilities for handling the most common types of nonlinearity.
Changing status or contact nonlinearity —Many common structural features exhibit nonlinearbehavior that is status-dependent. Status changesmight be directly related to load, or they might bedetermined by some external cause. Situations inwhich contact occurs are common to many differentnonlinear applications. Contact behavior, such as separation and sliding with frictional effects, introduces nonlinearity into the analysis.
Geometric nonlinearity — If a structure experiences large deformations, its changing geometric configuration can cause the structure to respond nonlinearly. Geometric nonlinearity is
characterized by “large” displacements and/or rotations. Small deflection and small strain analysisassume that displacements are small enough that the resulting stiffness changes are insignificant. In contrast, large strain analysis account for the stiffnesschanges that result from changes in an element’sshape and orientation. The large strain feature is available in most of the solid elements (including all of the large strain elements) as well as in most of the shell and beam elements. ANSYS also handlestwo other types of geometric nonlinearities: stressstiffening and spin softening.
For thin, highly stressed structures, such ascables and membranes, the out-of-plane stiffness of astructure can be affected significantly by the state ofin-plane stress in that structure. Stress stiffness is thecoupling between in-plane stress and transverse stiffness. Spin softening softens the stiffness matrix ofa rotating body for dynamic mass effects. The adjust-ment approximates the effects of geometry changes
due to large deflection circumferential motionin a small deflection analysis. Spin softeningis used in conjunction with prestressing,which is caused by centrifugal force in therotating body.
Material nonlinearity — Nonlinearstress–strain relationships are a commoncause of nonlinear structural behavior. Many factors can influence a material’sstress–strain properties, including load history (as in elastoplastic response), environmental conditions (such as temperature) and the amount of time that aload is applied (as in creep response).ANSYS handles numerous material-relatedfactors that cause a structure’s stiffness tochange during the course of an analysisranging from anisotropic behavior, nonlinearstress–strain relationships, dependency ontime, rate of strain and certain coupledphysics effects such as piezoelectric andSeebeck effects, to name a few.
By Achuth Rao, Ph.D.Product ManagerANSYS, Inc.
Nonlinear history tracking option monitors results in real time during solution.
ANSYS Nonlinear Technology
Powerful new capabilities are aimed at studyingcomplex nonlinear behavior in mechanical systems.
ANSYS employs the Newton-Raphson technique tosolve the previously mentioned types of nonlinearities,in which the out-of-balance load (the differencebetween the restoring forces and the applied loads) isused to perform a linear solution. ANSYS checks for convergence based on force, displacement or other criteria. If convergence criteria are not satisfied, the stiffness matrix is updated and a new solution is obtained.
A number of convergence-enhancement andrecovery features are offered by default such as line search, automatic load stepping and bisection. For special cases such as nonlinear buckling, ANSYS offers an alternative iteration scheme, the arc-length method, to help avoid bifurcation points and track unloading.
Latest ANSYS Capabilities
Recent releases of ANSYS have seen further advancesin nonlinearity and solution techniques for handlingthese types of nonlinear behavior.
Manual rezoning — In a finite large-deformationanalysis, mesh distortion reduces simulation accuracy,causes convergence difficulties and eventually can terminate an analysis. Rezoning allows you to repairthe distorted mesh and continue the simulation.ANSYS offers a manual rezoning procedure that allowsusers to decide when to use rezoning and whatregion(s) to rezone, and then to generate a new meshon the selected region(s). During the rezoning process,ANSYS updates the database as necessary, generatescontact elements if needed, transfers boundary conditions and loads from the original mesh and mapsall solved variables (node and element solutions) to thenew mesh automatically. Analysis then continues onthe new mesh, with equilibrium achieved based on themapped variables.
Nonlinear diagnostics — The nonlinear diagnosticstool in ANSYS can help you find problems in your modelwhen a nonlinear analysis has difficulty converging. Typically, nonlinear analysis fail to converge for the following reasons:
■ Too large a distortion
■ Elements contain nodes that have near-zero pivots (nonlinear analysis)
■ Too large a plastic or creep strain increment
■ Elements in which mixed u-P constraints are not satisfied
Tracking nonlinear residuals — As part of the nonlinear diagnostics, ANSYS allows tracking of the Newton-Raphson residuals during nonlinear iterations.Plotting the residual forces helps identify regions of highresidual forces. Such a capability is useful when you experience convergence difficulties in the middle of a loadstep, in which the model has a large number of contactsurfaces and other nonlinearities. Tracking the nonlinearresiduals allows one to focus on the nonlinearities in areaof interest, instead of having to deal with the entire model.Nonlinear diagnostics also allows one to identify elementsthat violate certain convergence criteria, such asplastic/creep strain increments and the like. The nonlinearhistory tracking option allows one to monitor results ofinterest in real time during solution. Before starting thesolution, you can request nodal data, such as displace-ments or reaction forces at specific nodes. You also canrequest element nodal data, such as stresses and strainsat specific elements, to be graphed.
Brake squeal analysis — The QR damped eigenvalue extraction method now can be used in problems with friction nonlinearities, in which an unsym-metric stiffness matrix may be produced. An example ofthis type of problem is brake squeal analysis, in which thecombination of ANSYS contact elements and theQRDAMP eigensolver provide an easy-to-use, efficient
Plotting Newton-Raphson residuals allows users to readily evaluate convergence difficulties.
means of determining unstable modes. ANSYS offersa two-step procedure in which the nonlinear unsymmetric stiffness terms due to frictional sliding ina static analysis are included in the eigensolution. Inbrake squeal analysis, the effect of the coefficient offriction (as well as other parameters) can be varied tosee the effects on different modes and the couplingbetween modes. This can help to determine whichmodes (frequencies) will be unstable and a source ofaudible discomfort.
Coupled physics — Due to interaction of variousphysics, coupled physics analysis is inherently nonlinear in nature. The interaction between variousphysics is typically either as a load or as a change inthe stiffness of the other physics. This type of inter-action makes the coupled system of equations nonlinear. ANSYS offers two types of coupled physicscapabilities: direct coupled physics and sequentialcoupled physics.
The direct method usually involves just oneanalysis that uses a coupled-field element type containing all necessary degrees of freedom. Couplingis handled by calculating element matrices or elementload vectors that contain all necessary terms. Anexample of this is a coupled physics analysis using thePLANE223, SOLID226 or SOLID227 elements. Userscan define material properties for these elements tomodel interaction such as piezoelectric, piezoresistive,Seebeck/Peltier effects and the piezocaloric effect.
The sequential method involves two or moresequential analysis, each belonging to a different field.The ANSYS Multi-field solver, available for a largeclass of coupled analysis problems, is an automatedtool for solving sequentially coupled field problems. It is built on the premise that each physics is createdas a field with an independent solid model and mesh. Coupled loads automatically are transferredacross dissimilar meshes by the solver. The solver isapplicable to static, harmonic and transient analysis,depending on the physics requirements. Any numberof fields may be solved in a sequential (or mixedsequential/simultaneous) manner. An application ofthe ANSYS Multi-field solver (MFX-Multiple codesolver) used for simulations with physics fields distributed between more than one product executable is the ANSYS Multiphysics and ANSYSCFX coupling for advanced FSI analysis. The solveruses iterative coupling in which each physics is solvedeither simultaneously or sequentially, and each matrixequation is solved separately. The solver iteratesbetween each physics field until loads transferredacross the physics interfaces converge.
In addition to some of the recent advances mentioned in this article, ANSYS continues toenhance its nonlinear capability. The next version ofANSYS will have further advances in areas of contactnonlinearity (line-surface contact, cohesive zonemodel using contact elements), material nonlinearity(Gurson’s material, anisotropic hyperelasticity), element or geometric nonlinearity (higher order shell,rebar elements) and convergence enhancementtechniques (stabilization). �
The author wishes to thank development and technical support personnel at ANSYS, Inc. and the various third-party solutions providers for their efforts and contribution tothis article.
Sequential analysis between ANSYS CFX and ANSYS Multiphysics provides for nonlinear coupled physics analysis of a MEMS micro-pump.
