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\left[ \widetilde { K } _ { i } ^ { N M } \right] ^ { - 1 } 1 using a "product plus increment" form:
Equation 2.2.2-3
\left[ \widetilde {K} _ {i} ^ {M N} \right] ^ {- 1} = \left[ I ^ {N L} - \rho_ {i} c _ {i} ^ {N} \gamma_ {i} ^ {L} \right] \left[ \widetilde {K} _ {i - 1} ^ {P L} \right] ^ {- 1} \left[ I ^ {P M} - \rho_ {i} \gamma_ {i} ^ {P} c _ {i} ^ {M} \right] + \rho_ {i} c _ {i} ^ {N} c _ {i} ^ {M},
where
\rho_ {i} = \left(c _ {i} ^ {M} \gamma_ {i} ^ {M}\right) ^ {- 1}.
In the actual implementation of this version of the BFGS method, each \left[ \widetilde { K } _ { i } ^ { M N } \right] ^ { - 1 } is not stored: rather, a "kernel" matrix, \left[ \widetilde { K } _ { I } ^ { M N } \right] ^ { - 1 } , is used (as the decomposition of { \widetilde K } _ { I } ^ { M N } ) , and the update is accomplished by premultiplication of the kernel matrix by the terms
\left[ I ^ {N L} - \rho_ {j} c _ {j} ^ {N} \gamma_ {j} ^ {L} \right]
and postmultiplication of the kernel matrix by the terms
\left[ I ^ {P M} - \rho_ {j} \gamma_ {j} ^ {P} c _ {j} ^ {M} \right]
for j = I + 1 , I + 2 , \ldots i . Because of the form of these terms, the premultiplication and postmultiplication operations result in inner products of vectors and the scaling of vectors by constants: it is this organization that makes the method computationally attractive. However, too many such products ( i - I being bigger than, say, 5-10) are not attractive, so usually a new kernel matrix is formed and stored after some iterations. In the ABAQUS/Standard implementation the kernel is the actual Jacobian matrix \partial F ^ { N } / \partial u ^ { M } . It is formed whenever a specified number of iterations have been done without obtaining a convergent solution: the parameter REFORM KERNEL on the *SOLUTION TECHNIQUE option determines the number of such iterations. A default of 8 iterations is provided for this parameter. ABAQUS/Standard does not reform the kernel unless this value is exceeded, so the same kernel can be used for several increments if the BFGS updates are successful.
In general, the rate of convergence of the quasi-Newton method is slower than the quadratic rate of convergence of Newton's method, though faster than the linear rate of convergence of the modified Newton method.
2.3 Buckling and postbuckling
2.3.1 Eigenvalue buckling prediction
ABAQUS/Standard contains a capability for estimating elastic buckling by eigenvalue extraction. This estimation is typically useful for "stiff" structures, where the prebuckling response is almost linear. The buckling load estimate is obtained as a multiplier of the pattern of perturbation loads, which are added to a set of base state loads. The base state of the structure may have resulted from any type of response
Procedures
history, including nonlinear effects. It represents the initial state to which the perturbation loads are added. The response to the perturbation loads must be elastic up to the estimated buckling load values for the eigenvalue estimates to be reasonable.
The following physical problem is addressed in eigenvalue buckling analysis: from an arbitrarily achieved base configuration with stresses { \pmb \sigma } ^ { B } in equilibrium with surface traction \mathbf { t } ^ { B } and body forces \mathbf { q } ^ { B } , we consider an elastic deformation with "small" displacement gradients under additional surface tractions \Delta \mathbf { t } , body forces \Delta \mathbf { b } _ { \mathrm { : } } , and boundary displacements \Delta \mathbf { u } . , where the additional tractions and displacements are applied on mutually complementary parts of the boundary. Such a deformation is a linear perturbation on a predeformed state. A consistent application of the small-displacement gradient assumption to the kinematics and the constitutive equation from an initially stressed state leads to the solution of a linear problem as the response to the additional loading. Since the problem is linear, if \Delta \pmb { \sigma } is the stress response to the loads \Delta \mathbf { t } , \Delta \mathbf { q } . , and \Delta \mathbf { u } , then for loads \lambda \Delta \mathbf { t } , \lambda \Delta \mathbf { q } , and \lambda \Delta \mathbf { u } the stress response will be \lambda \Delta \pmb { \sigma } .
Each distinct value of ¸ corresponds to a linear perturbation of the base state. Among these perturbed states we seek special values of ¸ that allow for the existence of nontrivial incremental displacement fields with arbitrary magnitudes as valid solutions to the problem. Such nontrivial incremental displacement fields are referred to as buckling modes. In the buckling analysis procedure in ABAQUS we do not distinguish between the geometry of the base state and the linearly perturbed configurations. As a result of this assumption we can seek the buckling modes as incremental displacements out of the base state geometry with stresses \pmb { \sigma } ^ { B } + \lambda \Delta \pmb { \sigma } , applied tractions \mathbf { t } ^ { B } + \lambda \Delta \mathbf { t } , and applied body forces \mathbf { q } ^ { B } + \lambda \Delta \mathbf { q } .
The equations of equilibrium for an arbitrarily chosen configuration during buckling, referred to as the current configuration, are written in terms of the nominal stress P in the base state. If X represents the position of a material point in the base state, the equilibrium equations can be expressed as
\int_ {V ^ {B}} \mathbf {P}: \frac {\partial \bar {\mathbf {v}}}{\partial \mathbf {X}} d V ^ {B} = \int_ {S ^ {B}} \mathbf {p} \cdot \bar {\mathbf {v}} d S ^ {B} + \int_ {V ^ {B}} \mathbf {b} \cdot \bar {\mathbf {v}} d V ^ {B},
where v¹ is an arbitrary virtual velocity field, p is the nominal traction on the boundary S ^ { B } of the body in the base state, b represents the body force per unit volume in the base state, and V ^ { B } is the volume that the body occupies in the base state. The corresponding rate form is given by
Equation 2.3.1-1
\int_ {V ^ {B}} \dot {\bf P}: \frac {\partial \bar {\bf v}}{\partial {\bf X}} d V ^ {B} = \int_ {S ^ {B}} \dot {\bf p} \cdot \bar {\bf v} d S ^ {B} + \int_ {V ^ {B}} \dot {\bf b} \cdot \bar {\bf v} d V ^ {B}.
Since we have assumed that the base state and the current state are indistinguishable, we now proceed to express the left-hand side in terms of the rate of Kirchhoff stress { \dot { \pmb { \tau } } } , the velocity gradient \mathbf { L } , the virtual velocity gradient L¹, and the deformation gradient F. Using the relations \mathbf { P } = \mathbf { F } ^ { - 1 } \cdot \boldsymbol { \tau } , where ¿ is the Kirchhoff stress based on the base state as the reference configuration, and \dot { \bar { \mathbf { F } } } = \bar { \mathbf { L } } \cdot \mathbf { F } , Equation 2.3.1-1 takes the form
Procedures
\int_ {V ^ {B}} \dot {\bf P}: \frac {\partial \bar {\bf v}}{\partial {\bf X}} d V ^ {B} = \int_ {V ^ {B}} \left[ \dot {\pmb {\tau}}: \bar {\bf L} - (\pmb {\tau} \cdot \bar {\bf L}): {\bf L} \right] d V ^ {B}.
We now use the relation between the rate of Kirchhoff stress { \dot { \pmb { \tau } } } , the material spin \begin{array} { r } { \pmb { \omega } = \frac { 1 } { 2 } ( \mathbf { L } - \mathbf { L } ^ { T } ) } \end{array} , and the Jaumann rate of Kirchhoff stress \tau ^ { \nabla } to transform this expression into
\int_ {V ^ {B}} \dot {\bf P}: \frac {\partial \bar {\bf v}}{\partial {\bf X}} d V ^ {B} = \int_ {V ^ {B}} \left[ {\pmb \tau} ^ {\nabla}: \bar {\bf D} + {\pmb \tau}: (\bar {\bf L} ^ {T} \cdot {\bf L} - 2 {\bf D} \cdot \bar {\bf D}) \right] d V ^ {B}.
In addition, we can replace the Kirchhoff stress ¿ with the Cauchy stress ¾ since it is assumed that the current and reference configurations are indistinguishable.
For the right-hand side of Equation 2.3.1-1 we note that the nominal tractions p and body forces b are given by \mathbf { p } = \mathbf { t } d S / d S ^ { B } and \mathbf { b } = \mathbf { q } d V / d V ^ { B } , where d S and d V are the elements of surface area and volume in the current configuration. For any material point the changes in t and \mathbf { q } during buckling are completely characterized by the change of the deformation gradient at that point; loosely speaking, the magnitude of the applied forces at any material point is kept fixed, and the change in the applied tractions and body force intensities arises due to the change in geometry. For example, for a pressure load the magnitude of the pressure remains constant but the surface normal changes--a change that is completely characterized by the change in the deformation gradient. Since the ratios of the surface area and volume measures between the reference and current configurations can be viewed as functions of the deformation gradient F only, it follows that p and b at any given material point also change only through their dependence on the deformation gradient; hence, their rates of change can be written as
\dot {\mathbf {p}} = \frac {\partial \mathbf {p}}{\partial \mathbf {F}}: \dot {\mathbf {F}} \qquad \mathrm{and} \qquad \dot {\mathbf {b}} = \frac {\partial \mathbf {b}}{\partial \mathbf {F}}: \dot {\mathbf {F}},
or when the current and reference configurations are indistinguishable,
\dot {\mathbf {p}} = \frac {\partial \mathbf {p}}{\partial \mathbf {F}}: \mathbf {L} \quad \text {and} \quad \dot {\mathbf {b}} = \frac {\partial \mathbf {b}}{\partial \mathbf {F}}: \mathbf {L}.
Assuming a hypoelastic constitutive law,
\boldsymbol {\tau} ^ {\nabla} = \mathbf {C} (\boldsymbol {\sigma}): \mathbf {D},
where \mathbf { C } ( \pmb { \sigma } ) is a fourth-order tensor that can depend on the current stress, the governing equation for the buckling analysis becomes
Equation 2.3.1-2
Procedures
\begin{array}{l} \int_ {V ^ {B}} \bar {\mathbf {D}}: \mathbf {C} (\pmb {\sigma} ^ {B}): \mathbf {D} d V ^ {B} + \int_ {V ^ {B}} (\pmb {\sigma} ^ {B} + \lambda \Delta \pmb {\sigma}): (\bar {\mathbf {L}} ^ {T} \cdot \mathbf {L} - 2 \mathbf {D} \cdot \bar {\mathbf {D}}) d V ^ {B} - \\ \int_ {S ^ {B}} \bar {\mathbf {v}} \cdot \left(\frac {\partial \mathbf {p} ^ {B}}{\partial \mathbf {F}}: \mathbf {L}\right) d S ^ {B} - \int_ {S ^ {B}} \bar {\mathbf {v}} \cdot \left(\frac {\partial \lambda \Delta \mathbf {p}}{\partial \mathbf {F}}: \mathbf {L}\right) d S ^ {B} - \\ \int_ {V ^ {B}} \bar {\mathbf {v}} \cdot \left(\frac {\partial \mathbf {b} ^ {B}}{\partial \mathbf {F}}: \mathbf {L}\right) d V ^ {B} - \int_ {V ^ {B}} \bar {\mathbf {v}} \cdot \left(\frac {\partial \lambda \Delta \mathbf {b}}{\partial \mathbf {F}}: \mathbf {L}\right) d V ^ {B} = \mathbf {0}, \\ \end{array}
where \mathbf { p } ^ { B } and \lambda \Delta \mathbf { p } are the nominal tractions generated during buckling corresponding to the base state tractions \mathbf { t } ^ { B } and the linear perturbation tractions \Delta \mathbf { t } , , respectively; similar definitions apply for the nominal body force terms. The constitutive relation can represent elasticity, hypoelasticity, and hyperelasticity; rate effects and plasticity are ignored. The effective moduli are evaluated for the value of the stress and deformation in the base state.
To derive the finite element discretization for the expression above, we introduce the interpolated velocity field
\mathbf {v} = v ^ {N} \mathbf {N} ^ {N} (\mathbf {X}),
where X represents the position in the base state. Using the standard finite element approach, the governing equations for buckling then take the form of the standard eigenvalue problem:
(K _ {0} ^ {N M} + \lambda K _ {\Delta} ^ {N M}) v ^ {M} = 0,
where K _ { 0 } ^ { N M } is the base state stiffness and K _ { \Delta } ^ { N M } is the differential stiffness. The base state stiffness is the sum of the hypoelastic tangent stiffness, the initial stress stiffness, and the load stiffness:
\begin{array}{l} K _ {0} ^ {N M} = \int_ {V ^ {B}} \left(\frac {\partial \mathbf {N} ^ {N}}{\partial \mathbf {x}}\right) _ {s y m}: \mathbf {C} (\pmb {\sigma} ^ {B}): \left(\frac {\partial \mathbf {N} ^ {M}}{\partial \mathbf {x}}\right) _ {s y m} d V ^ {B} + \\ \int_ {V ^ {B}} \pmb {\sigma} ^ {B}: \left[ \left(\frac {\partial \mathbf {N} ^ {N}}{\partial \mathbf {x}}\right) ^ {T} \cdot \frac {\partial \mathbf {N} ^ {M}}{\partial \mathbf {x}} - 2 \left(\frac {\partial \mathbf {N} ^ {M}}{\partial \mathbf {x}}\right) _ {s y m} \cdot \left(\frac {\partial \mathbf {N} ^ {N}}{\partial \mathbf {x}}\right) _ {s y m} \right] d V ^ {B} - \\ \int_ {S ^ {B}} \mathbf {N} ^ {N} \cdot \frac {\partial \mathbf {p} ^ {B}}{\partial u ^ {M}} d S ^ {B} - \int_ {V ^ {B}} \mathbf {N} ^ {N} \cdot \frac {\partial \mathbf {b} ^ {B}}{\partial u ^ {M}} d V ^ {B}, \\ \end{array}
where \partial \mathbf { p } ^ { B } / \partial u ^ { M } and \partial \mathbf { b } ^ { B } / \partial u ^ { M } are the derivatives of the nominal surface tractions and body forces with respect to the nodal displacements. Since we do not distinguish between the current configuration and the reference configuration, the partial derivatives appearing in the load stiffness terms are all evaluated at u ^ { M } = 0 corresponding to \mathbf { F } = \mathbf { I } . For example, the load stiffness term for the surface tractions appearing in Equation 2.3.1-2,
\bar {\mathbf {v}} \cdot \left(\frac {\partial \mathbf {p} ^ {B}}{\partial \mathbf {F}}: \mathbf {L}\right),
transforms into the finite element expression
\mathbf {N} ^ {N} \cdot \left(\frac {\partial \mathbf {p} ^ {B}}{\partial \mathbf {F}}: \frac {\partial \mathbf {N} ^ {M}}{\partial \mathbf {X}}\right) v ^ {M} = \mathbf {N} ^ {N} \cdot \frac {\partial \mathbf {p} ^ {B}}{\partial u ^ {M}} v ^ {M},
with
\mathbf {p} ^ {B} = \mathbf {p} ^ {B} \left(\mathbf {I} + \frac {\partial \mathbf {N} ^ {K}}{\partial \mathbf {X}} u ^ {K}\right).
The differential stiffness consists of the sum of the initial stress stiffness due to the perturbation stresses and the load stiffness due to the perturbation loads:
\begin{array}{l} K _ {\Delta} ^ {N M} = \int_ {V ^ {B}} \Delta \pmb {\sigma}: \left[ \left(\frac {\partial \mathbf {N} ^ {N}}{\partial \mathbf {x}}\right) ^ {T} \cdot \frac {\partial \mathbf {N} ^ {M}}{\partial \mathbf {x}} - 2 \left(\frac {\partial \mathbf {N} ^ {M}}{\partial \mathbf {x}}\right) _ {s y m} \cdot \left(\frac {\partial \mathbf {N} ^ {N}}{\partial \mathbf {x}}\right) _ {s y m} \right] d V ^ {B} - \\ \int_ {S ^ {B}} \mathbf {N} ^ {N} \cdot \frac {\partial \Delta \mathbf {p}}{\partial u ^ {M}} d S ^ {B} - \int_ {V ^ {B}} \mathbf {N} ^ {N} \cdot \frac {\partial \Delta \mathbf {b}}{\partial u ^ {M}} d V ^ {B}. \\ \end{array}
The contribution in this expression that is derived from the stress is symmetric; however, the contribution derived from the applied loads (the load stiffness) is symmetric only if the applied loading is conservative--that is, if the loads can be derived from an energy potential. If the load stiffness is nonsymmetric, the contribution will be symmetrized since ABAQUS can solve eigenvalue problems only with symmetric matrices.
If the generalized nodal "loads" resulting from both applied forces \mathbf { t } ^ { B } and \mathbf { q } ^ { B } as well as prescribed displacements \mathbf { u } ^ { B } are denoted by P ^ { N } and those due to \Delta \mathbf { t } , \Delta \mathbf { q } . , and \Delta \mathbf { u } are denoted by Q ^ { N } , the eigenvalues \lambda _ { i } represent the multipliers that provide the estimated generalized buckling load as P ^ { N } + \lambda _ { i } Q ^ { N } , while the corresponding eigenvectors v _ { i } ^ { N } give the associated buckling modes. Although in most analyses the lowest mode is the only one of interest, ABAQUS is able to extract several modes simultaneously. It is also worth noting that the common case of an antisymmetric buckling mode on a symmetric base state and buckling load is easily done with ABAQUS--refer to ``Eigenvalue buckling prediction,'' Section 6.2.3 of the ABAQUS/Standard User's Manual, for details.
If the tangent stiffness is predicted poorly by K _ { 0 } ^ { N M } + \lambda K _ { \Delta } ^ { N M } (that is, the structure is not "stiff" in the sense that the response is nonlinear prior to buckling), a nonlinear analysis using the Riks method is required to obtain a reliable estimate for the load carrying capacity of the structure.
2.3.2 Modified Riks algorithm
It is often necessary to obtain nonlinear static equilibrium solutions for unstable problems, where the load-displacement response can exhibit the type of behavior sketched in Figure 2.3.2-1--that is, during periods of the response, the load and/or the displacement may decrease as the solution evolves. The modified Riks method is an algorithm that allows effective solution of such cases.
Figure 2.3.2-1 Typical unstable static response.
text_image
Load load maximum displacement maximum displacement minimum load minimum Displacement
It is assumed that the loading is proportional--that is, that all load magnitudes vary with a single scalar parameter. Also, we assume that the response is reasonably smooth--that sudden bifurcations do not occur. Several methods have been proposed and applied to such problems. Of these, the most successful seems to be the modified Riks method--see, for example, Crisfield (1981), Ramm (1981), and Powell and Simons (1981)--and a version of this method has been implemented in ABAQUS. The essence of the method is that the solution is viewed as the discovery of a single equilibrium path in a space defined by the nodal variables and the loading parameter. Development of the solution requires that we traverse this path as far as required. The basic algorithm remains the Newton method; therefore, at any time there will be a finite radius of convergence. Further, many of the materials (and possibly loadings) of interest will have path-dependent response. For these reasons, it is essential to limit the increment size. In the modified Riks algorithm, as it is implemented in ABAQUS, the increment size is limited by moving a given distance (determined by the standard, convergence rate-dependent, automatic incrementation algorithm for static case in ABAQUS/Standard) along the tangent line to the current solution point and then searching for equilibrium in the plane that passes through the point thus obtained and that is orthogonal to the same tangent line. Here the geometry referred to is the space of displacements, rotations, and the load parameter mentioned above.
Basic variable definitions
Let P ^ { N } \left( N = 1 , 2 , \dots = \right. the degrees of freedom of the model) be the loading pattern, as defined on the *CLOAD, *DLOAD, etc., options. Let ¸ be the load magnitude parameter, so at any time the actual load state is \lambda P ^ { N } , and let u ^ { N } be the displacements at that time.
The solution space is scaled to make the dimensions approximately the same magnitude on each axis. In ABAQUS this is done by measuring the maximum absolute value of all displacement variables, u, in the initial (linear) iteration. We also define \overline { { P } } = ( P ^ { N } P ^ { N } ) ^ { \frac { 1 } { 2 } } . The scaled space is then spanned by
Procedures
\mathbf { l o a d } = \lambda \tilde { P } ^ { N } , \ \tilde { P } ^ { N } = P ^ { N } / \overline { { P } } ,
displacements \mathbf { \Psi } = \tilde { u } ^ { N } = ( u ^ { N } / \overline { { u } } ) ;
and the solution path is then the continuous set of equilibrium points described by the vector ( \tilde { u } ^ { N } ; \lambda ) in this scaled space. All components of this vector will be of order unity. The algorithm is shown in Figure 2.3.2-2 and is described below.
Figure 2.3.2-2 Modified Riks algorithm.
text_image
λ A¹ A² (¯v¹ᴺ ;1) ¯(ṽ⁰ᴺ ;1) ρ₁ Equilibrium surface A⁰ ¯uᴺ
Suppose the solution has been developed to the point A ^ { o } = ( \tilde { u } _ { o } ^ { N } ; ~ \lambda _ { o } ) \nonumber . The tangent stiffness, K _ { o } ^ { N M } , is formed, and we solve
K _ {o} ^ {N M} v _ {o} ^ {M} = P ^ {N}.
The increment size ( A ^ { o } to A ^ { 1 } in Figure 2.3.2-2) is chosen from a specified path length, \Delta l . , in the solution space, so that
\Delta \lambda_ {o} ^ {2} (\tilde {v} _ {o} ^ {N}; 1): (\tilde {v} _ {o} ^ {N}; 1) = \Delta l ^ {2},
and, hence,
\Delta \lambda_ {o} = \frac {\pm \Delta l}{\left(\tilde {v} _ {o} ^ {N} \tilde {v} _ {o} ^ {N} + 1\right) ^ {\frac {1}{2}}}
(here \tilde { v } _ { o } ^ { N } is v _ { o } ^ { N } scaled by u). The value \Delta l is initially suggested by the user and is adjusted by the ABAQUS/Standard automatic load incrementation algorithm for static problems, based on the convergence rate. The sign of \Delta \lambda _ { o } --the direction of response along the tangent line--is chosen so that
Procedures
the dot product of \Delta \lambda _ { o } ( \tilde { v } _ { o } ^ { N } ; \ 1 ) on the solution to the previous increment, ( \Delta \tilde { u } _ { - 1 } ^ { N } ; \ \Delta \lambda _ { - 1 } ) , is positive:
\Delta \lambda_ {o} (\tilde {v} _ {o} ^ {N}; 1): (\Delta \tilde {u} _ {- 1} ^ {N}; \Delta \lambda_ {- 1}) > 0,
that is
\Delta \lambda_ {o} (\tilde {v} _ {o} ^ {N} \Delta \tilde {u} _ {- 1} ^ {N} + \Delta \lambda_ {- 1}) > 0.
It is possible that in some cases, where the response shows very high curvature in the ( \tilde { u } ^ { N } ; \lambda ) space, this criterion will cause the wrong sign to be chosen--see, for example, Figure 2.3.2-3.
Figure 2.3.2-3 Example of incorrect choice of sign for \Delta \lambda .
text_image
(Δ û̃_1^N ; Δλ̃_1) λ A⁰ This direction will be chosen and will take the solution the wrong way along the equilibrium path. û ~
The wrong sign is rarely chosen in practical cases, unless the increment size is too large or the solution bifurcates sharply. To check for such cases is computationally expensive: one approach would be for the solution to be found at \lambda _ { o } - \varepsilon \Delta \lambda _ { - 1 } , 0 < \varepsilon < < 1 , so that we obtain a vector that gives a close approximation of the directed tangent at A ^ { o } . Because the case is so rare, such a check is not included, and the simple dot product given above is used alone to determine the sign of \Delta \lambda ^ { o } . Thus, we have now found the point A ^ { 1 } \left( \tilde { u } _ { o } ^ { N } + \Delta \lambda _ { o } \tilde { v } _ { o } ^ { N } ; \lambda _ { o } + \Delta \lambda _ { o } \right) in Figure 2.3.2-2. The solution is now corrected onto the equilibrium path in the plane passing through A ^ { 1 } and orthogonal to ( \tilde { v } _ { o } ^ { N } ; 1 ) , by the following iterative algorithm.
Initialize: \Delta \lambda _ { i } = \Delta \lambda _ { o } , \ \Delta u _ { i } ^ { N } = \Delta \lambda _ { 0 } v _ { o } ^ { N }
For i = iteration (i = 1; 2; 3; etc:):
a. Form I ^ { N } , \ K ^ { N M } ; the internal (stress) forces at the nodes,
I ^ {N} = \int_ {V} \pmb {\beta} ^ {N}: \pmb {\sigma} d V, \mathrm{and} K ^ {N M} = \frac {\partial I ^ {N}}{\partial u ^ {M}},
at the state ( u _ { o } ^ { N } + \Delta u _ { i } ^ { N } ; \lambda _ { o } + \Delta \lambda _ { i } ) --that is, at A ^ { i } in Figure 2.3.2-2.
Procedures
b. Check equilibrium:
R _ {i} ^ {N} = (\lambda_ {o} + \Delta \lambda_ {i}) P ^ {N} - I ^ {N}.
If all the entries in R _ { i } ^ { N } are sufficiently small, the increment has converged. If not, we proceed.
c. Solve:
K ^ {N M} \left\{v _ {i} ^ {M}; c _ {i} ^ {M} \right\} = \left\{P ^ {N}; R _ {i} ^ {N} \right\}.
That is, we solve simultaneously with two load vectors, P ^ { N } and R ^ { N } , and obtain two displacement vectors, v _ { i } ^ { N } and c _ { i } ^ { N } .
d. Now scale the vector ( \tilde { v } _ { i } ^ { N } ; 1 ) , and add it to \left( \tilde { c } _ { i } ^ { N } ; \rho _ { i } \right) where \rho _ { i } = R _ { i } ^ { N } P ^ { N } / \overline { { P } } ^ { 2 } is the projection of the scaled residuals onto \tilde { P } ^ { N } so that we move from A ^ { i } to A ^ { i + 1 } in the plane orthogonal to ( \tilde { v } _ { o } ^ { N } ; 1 ) --see Figure 2.3.2-2. This gives the equation
\left\{(0; - \rho_ {i}) + (\tilde {c} _ {i} ^ {N}; \rho_ {i}) + \mu (\tilde {v} _ {i} ^ {N}; 1) \right\}: (\tilde {v} _ {o} ^ {N}; 1) = 0,
which simplifies to give
\mu = - \frac {\tilde {c} _ {i} ^ {N} \tilde {v} _ {o} ^ {N}}{\tilde {v} _ {i} ^ {N} \tilde {v} _ {o} ^ {N} + 1},
and the solution point is now A ^ { i } :
(u _ {o} ^ {N} + \Delta u _ {i} ^ {N} + c _ {i} ^ {N} + \mu v _ {i} ^ {N}; \lambda_ {o} + \Delta \lambda_ {i} + \mu).
e. Update for the next iteration,
\Delta u _ {i + 1} ^ {N} = \Delta u _ {i} ^ {N} + c _ {i} ^ {N} + \mu v _ {i} ^ {N}
\Delta \lambda_ {i + 1} = \Delta \lambda_ {i} + \mu
i = i + 1,
and return to (a) above for the next iteration.
The implementation in ABAQUS/Standard includes the additional update after each iteration:
v _ {o} ^ {N} = v _ {i} ^ {N}.
This causes the equilibrium search to be orthogonal to the last tangent, rather than to the tangent at the beginning of the increment. The main motivation for this additional modification comes from the use
of the method in plasticity problems, where the first iteration of each increment uses the elastic material stiffness to establish the direction of straining and so provides a stiffness that is not representative of the tangent to the equilibrium path if active plasticity is occurring.
The total path length traversed is determined by the load magnitudes supplied by the user on the loading options, while the number of increments is determined by the data line supplied with the *STATIC option, assisted by ABAQUS/Standard's automatic incrementation scheme if that is chosen.
2.4 Nonlinear dynamics
2.4.1 Implicit dynamic analysis
ABAQUS offers dynamic analysis options for both linear and nonlinear problems. In the case of purely linear systems methods based on the eigenmodes of the system are almost always chosen, because they can provide insight into the structure's behavior that is not otherwise available and because they are usually significantly more cost-effective than the direct integration methods that are usually used for nonlinear problems. The linear dynamic analysis methods provided in ABAQUS/Standard are discussed in ``Modal dynamics,'' Section 2.5. For mildly nonlinear dynamic analysis problems the "modal projection method" is provided. The basis of that method is to use the eigenmodes of the linear system (extracted with the *FREQUENCY option) as a set of global Ritz functions--a set of global interpolation functions, in the terminology of the finite element method--whose amplitudes define the response. ABAQUS/Standard provides direct time integration using the explicit, central difference operator for this option. For any more severely nonlinear case the dynamic response is obtained by direct time integration of all of the degrees of freedom of the finite element model. The methods provided for this type of analysis are discussed in this section.
The choice of operator used to integrate the equations of motion in a dynamic analysis is influenced by many factors. ABAQUS/Standard is designed to analyze structural components, by which we mean that the overall dynamic response of a structure is sought, in contrast to wave propagation solutions associated with relatively local response in continua. Belytschko (1976) labels these "inertial problems" and classifies them by stating that "wave effects such as focusing, reflection, and diffraction are not important." Structural problems are considered "inertial" because the response time sought is long compared to the time required for waves to traverse the structure.
Dynamic integration operators are broadly characterized as implicit or explicit. Explicit schemes, as used in ABAQUS/Explicit, obtain values for dynamic quantities at t + \Delta t based entirely on available values at time t. The central difference operator, which is the most commonly used explicit operator for stress analysis applications, is only conditionally stable, the stability limit being approximately equal to the time for an elastic wave to cross the smallest element dimension in the model. Implicit schemes remove this upper bound on time step size by solving for dynamic quantities at time t + \Delta t based not only on values at t , but also on these same quantities at t + \Delta t . But because they are implicit, nonlinear equations must be solved. In structural problems implicit integration schemes usually give acceptable solutions with time steps typically one or two orders of magnitude larger than the stability limit of simple explicit schemes, but the response prediction will deteriorate as the time step size increases relative to the period of typical modes of response. See, for example, Hilber, Hughes and Taylor (1978) for a discussion of such errors. Thus, the relative economy of the two