SDT-nlsim         Contents     Functions              PDF Index

1.1  Kinematics of non-linear systems

• Observation of strains, stresses, application of forces
• Generalized non-linear springs
• Generalized non-linear surfaces
• Non-linear volumes
• Kinematic reduction, observation and hyperreduction
• Manual definition of input and output (deprecated)

1.1.1  Observation of strains, stresses, application of forces

One is interested in solving equations of the general form

(1.1)

with q the finite element DOFs, M,C,K the mass, viscous damping, and stiffness matrices respectively, F_ext the external forces and F_NL the non-linear forces internal to the system which may depend on the FEM motion described by its DOFs q and their velocities q as well as possibly internal states of the non-linear elements q_NL. When internal states are necessary (friction, plasticity, ...), additional equations are provided to model their evolution.

Estimation of the non-linear forces can,, in a very general fashion, be decomposed in three steps: observation of strains, evaluation of constitutive law at a material point to compute stresses, application of stresses on the model as detailed below.

• Observation of non-linear strains is the first step

  (1.2)

  where strains may represent quantities dependent on the model displacement (relative motion, mechanical strain, ...), but also possibly internal states of the non-linearity if those are included in the definition of q. For HBM solutions q_NL will be assumed to be part of q but for transient simulations this may not be optimal.

  Strategies for the construction of the observation matrix c will be discussed for point to point connection in section ??, surfaces in section ??, volumes in section ??.

  In the time domain, generalized strains є (noted .unl in the code) are obtained by computing

  (1.3)

  In the implementation, the strain vector may have N_E components e_i, and strains at N_G material points (Gauss or physical points) may be stored as a single vector to allow vectorization of non-linearities of a given kind, thus leading to e_i,gk components.

  For HBM computations N_T times will be evaluated and stored as columns. The internal storage in field NL.unl is thus of the form

  (1.4)

  In a similar fashion strain rates are obtained using

  (1.5)

  and stored in field NL.vnl.

• From the observed strains, a constitutive law is used to estimate stresses (or generalized forces), as will be discussed in section ?? from data present in the NL.Fu field,

  (1.6)

  The generic representation of non-linearities should verify classical assumptions on objectivity and on the validity of constitutive relations, which is compatible with the idea that generalized strains are defined at a material point.

  The definition of a non-linear constitutive relation giving a definition of generalized stresses s_nl which has the same (N_E × N_C) × N_T size as u_nl and is thus stored NL.unl field to allow overwrite. This allows memory optimization even though the output result is a force and not a strain). When internal states are present, the strain field NL.unl may not be efficiently used to store stresses, it is then possible to store stresses in field NL.snl of size (N_S × N_G)× N_T) with N_S the number of stress components.

• Finally stresses can be applied on the model to obtain the model forces in the discretized model associated with DOFs q. That is

  (1.7)

  where the field NL.snl may be defined or, in cases with no internal states, it is possible to optimize memory by storing s_nl in NL.unl.

In the proposed framework,

• internal states of a non-linearity are expected be included within the generalized strains. For a time computation unl will thus be of size (N_E+N_int) × N_g (× N_T in the case of HBM computations). See section ?? for more details.
• It is generally useful to store current u_nl(t_n), initial u_nl0and previous u_nl(t_n−1) values as consecutive blocks in NL.unl(:,:,1),NL.unl(:,:,2),NL.unl(:,:,3).
• In many instances the observation c and command b matrices can be considered constant, but for contacts with large relative displacements they may need to be considered as operators depending on the model state and thus modified dynamically using C code.

1.1.2  Generalized non-linear springs

The simplest example is the observation of unaxial springs oriented along vector {d} where the generalized strain is the relative displacement

(1.8)

The non-linear implementation of the OpenFEM cbush element provides 6 generalized strains at a given location corresponding to relative translations and rotations. For EDID==-1 this is with respect to a local basis B=[x_e y_e z_e].

(1.9)

Initialisation of non-linear behavior in a cbush element group is performed when the NLdata property contains fields

• type='nl_inout' to let hbm_solve InitHBM build the needed observation matrix
• Fu='@UserFun' references a user function computing the non-linear force with the prototype call described in section ??
• .isens can be used to apply non-linearity in some directions only.

A sample problem with cbush can be found in section ??.

When considering non small angular rotations of the two nodes, the orientation basis in (??) cannot be considered considered constant. It is thus necessary to express motion of each part at the bushing node using large rotations

(1.10)

In non-linearities that implement such large rotation, SDT expects that the bushing frame is associated with the first node and that motion of the bushing node associated with the second. Thus the relative translation in first node frame is given by

(1.11)

For rotational springs at the bushing, there is no obvious large rotation formulation. The current implementation uses the difference of angles even though this does not always make sense.

1.1.3  Generalized non-linear surfaces

Zero thickness elements which provide 3 generalized strains which correspond at each gauss point to the relative motion of two surfaces in the normal and two tangential directions are implemented in p_zt. When an NLdata field is defined a vectorized non-linearity is initialized.

Contact elements, implemented in p_contact have an objective similar to that of zero thickness element but allow the handling of non-conform meshes. Subtype 1 does not account for large displacement so that matching can be performed once and kept constant during time. Subtype 2 allows for relative motion of two bodies (wheel moving on a rail for example) and currently only works for matching to triangular surfaces.

1.1.4  Non-linear volumes

SDT implements observation of strains in configurations where no geometry updating is needed using StressCut calls. This strategy is compatible with all physics (acoustics, piezo-electricity, ...) and can also be used for shells (to observe membrane strains and curvature) and beams (to observe axial elongation, shear, torsion and curvature).

For volumes, the choice of strain is obtained using the p_solid Isop parameters. For explicit mechanics in large deformation, the displacement gradient is used and obtained using Isop=100.

For generic implementation of multi-physic non-linear volumes, a generalized strain giving the displacement and its gradient for each field. For a field with 3 components at N nodes

(1.12)

As a an example, the standard linearized mechanical strain

(1.13)

can be related to the generic multi-physic strain using a transformation matrix T_Ep

(1.14)

The usual mechanical stiffness can thus be reformulated using B_MP

(1.15)

A generic non-linear multiphysic constitutive law is a function that will receive a NL structure with fields

• .unl a series of generic strain e_mp at each Gauss points thus a matrix with 4*Nf*Ng rows (4 components per field, repeated for each gauss point), and possibly multiple columns for different times.
• .vnl a series of strain velocities (time derivative of strain)
• .nodeG matrix of size Nfield * Ng containing interpolated fields (typically positions x,y,z,un1 but possibly also v1x,v1y,v1z for first orientation axis), .nodeEt is an int32 vector coding the name of each field.
• .MatType list of matrix types for which the tangent constitutive law is given.
• .constit contains the constant parameters read from model.pl.

From this information the function should return a structure with field

• .unl generic stresses a matrix with 4*Nf*Ng rows and Nt columns. The name should really be .fnl but the same field name .unl is used to allow memory reuse.
• .MatType propagated version of input field.
• .DD a matrix of size ones(4*Nf,4*Nf,Ng,length(NL.MatType)) giving the tangent constitutive law for each Gauss point and each desired matrix type. This corresponds to the T_Ep^T D T_Ep in (??).

1.1.5  Kinematic reduction, observation and hyperreduction

Once a kinematic reduction defined, where {q}=[T]{q_R} is defined, non-linear observation and command matrices are simply reduced using c_R=c*T and b_R=T^Tb.

For resultant observations, the strategy is however more complex. In a frequency domain model, the resultant is associated with Z_I(p,s) the dynamic stiffness restricted to an interface

(1.16)

so that the coefficients of the observation depends on frequencies and parameters involved in Z_I(p,s). In SDT, such linear combinations as described in section ??. Taking the case of a bushing where [Z_I]=E₍ω,a)[K_u], the resultant observer is simply given by [c_I][Z_I]=E₍ω,a)[c_I K_u].

When the observer is built around an operating condition where large displacement has occurred, the observation matrix may depend on the current position c(x). For example, the elongation of a large rotation rod is given by dl(t) = √{u₂−u₁}^T {u₂−u₁}−l₀, so that the linearized observation is given by

For the case of resultant observation in enforced displacement problems, R_I(t)=[T_I]^T F(t), the reduced equations of motion give an estimate of F_R(t) and not F_T, thus T_IR^T should be used when observing.

Hyperreduction is a strategy where kinematic reduction is combined with selection of a subset of non-linear integration points based on criteria on the work of reduced model. Thus from a set E of gauss points where each non-linear strain is described by u_nl^g=[c^g]{q}, one restricts to a subset E_HR. Needs more details.

1.1.6  Manual definition of input and output (deprecated)

While the general approach is to associate non-linearities with strains in elements as described in the following sections, it is possible to define a pro.NLdata structure giving

• .b,.c : manual command and observation definition. If the NLdata.DOF field is defined, placement of the observation matrix in done during the model assembly phase assuming .DOF correspond to mdof (projection with the Case.T matrix, c*T ,is done). If one want to define NL on active DOF, one will provide .adof field (rather than .DOF field): then c is assumed to be defined on active dof, and a simple placeindof is done. If there is no NLdata.DOF field, .c and .b matrices are assumed to be on mdof, and projected using Case.T. For storage during solves, see .b.
• .Sens,.Load Alternate form for .b, .c giving command and observation matrices as .Sens cell array of the form {SensType,SensData} where SensType is a string defining the sensor type and SensData a matrix with the sensor data (see sdtweb sensor). .Load data structure defining the command as a load (with .DOF and .def fields).