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1.1  Non-linear system representation

• Strains, stresses, application of forces
• Jacobian computations

1.1.1  Strains, stresses, application of forces

One is interested in solving equations of the general form

(1.1)

with

• q the finite element DOFs (full or reduced)
• M,C,K the mass, viscous damping, and stiffness matrices respectively
• F_ext the external forces with the sign convention that they are written on the right hand side
• F_NL the non-linear forces internal to the system (sign convention on the left hand side of the equation) which may depend on the FEM motion described by its generalized strains and possibly strain rates defined below.
• When internal states are necessary (friction, plasticity, ...), they are stored either as generalized strain components or DOF and 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 1.2.1, surfaces in section 1.2.2, volumes in section 1.2.3.

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

  (1.3)

  In a similar fashion, if the NL.vnl field exists strain rates are obtained using

  (1.4)

  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

  (1.5)

• From the observed strains, a constitutive law is used to estimate stresses (or generalized forces), as will be discussed in section 1.3 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 1.3.3 for more details.
• It is generally useful to store current u_nl(t_n), initial u_nl0 and 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  Jacobian computations

Many iterative schemes need derivatives of the non-linear forces with respect to strains (stiffness)

(1.8)

or strain rates (damping matrix)

(1.9)

See nl_spring NLJacobianUpdate for low level documentation and nl_solve TgtMdl for global calls. The derivatives at each integration point (often named Gauss point in the documentation) ∂ s_nl/∂ u_nl correspond to the material stiffness matrix called dd or Lambda in different parts of SDT. Since the b and c matrices combine all gauss points, the assembled matrix containing one block for each gauss point, is sometimes stored in a ddg field when individually modifying the gauss points is of interest (topology optimization applications for example).

When splitting of Jacobians is of interest xxx.