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A large number of user subroutines and utility routines is available in Abaqus, that are all programmed in Fortran. Subroutines are different for implicit (standard) and explicit solvers. Utility routines can be called inside of subroutines to take care of standard tasks. Here a small subset of the most important routines: 4
For the explicit solver an own set of user subroutines is available. The VUMAT is again the most important for us. 5
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Basic flow chart with data flow and actions from the start of the ABAQUS standard analysis until its end. 7
A more detailed flow of ABAQUS/Standard 8
The routine should be a Fortran source code or an object file *.obj. The simultaneous used of multiple subroutines has to be done in one file. If you restart, the user subroutine has to be assigned again. 9
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Forward integration (Forward Euler (explicit)) is a simple methods, however with stability problems. The strain increment has to be chosen small, much smaller than the magnitude of the elastic strain. Hence for explicit integration the time increment hast to be controlled. Implicit or semi implicit schemes have more complicated algorithms and local iterations might be needed, however they are in general more stable and converge. 13
Jacobian also called Functional matrix or derivative matrix is the m x mmatrix of all 1st partial derivatives of differentiable function. 14
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Flow direction=(deviatoric stress alpha)/predicted stress = flowrule Consistency condition 30
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The Newton-Raphson method is a recursive method, since iterations need the results of previous iterations. The fundamental idea is to linearized the function in a point by taking its tangent. The null of the tangent is then taken as an improved approximation of the null of the function. The obtained approximation is then taken as the starting point for the next iteration, until the approximation goes under a certain limit. The iteration method converges in the best case with quadratic convergence. The number of correct decimals then doubles in each iteration. Formally said, the following procedure is repeated until a desired accuracy is met: Since solving the delta x_n with the inverse of a matrix with consecutive multiplication with f(x) is costly and numerically unfavorable, the linear equation system JDx= f is solved instead, followed by x_n+1.. To solve the system several methods exist. If the Jacobian is invertible at the null and continuous, the method converges locally quadratic, what makes it quiet useful. 37
However there are also other methods around that do the job much better. Newton Raphson method: In each iteration, the tangent matrix has to be calculated freshly and a linear system of equations needs to be solved. Modified Netwon Raphson method: Here the tangent stiffness matrix is only calculated at the beginning of the iteration. All consecutive iterations use the same matrix, what gave the method the name of the method of initial stiffness's. Since many iterations are needed due to low convergence rates, it is mainly used for small non linearity. Quasi Newton methods are a class of numerical methods for solving non linear minimization problems. The methods are based on the Newton method, but calculate the inverse of the Hesse matrix not directly, but approximate them only to limit the computational cost per iteration. The first algorithm was proposes in 1950 by the physicist Davidon. The most known variants are the Broyden Fletcher Goldfarb Shanno (BFGS) and Davidon Fletcher Powell (DFP), Broyden or SR1 a.s.o. 38
The arc length method helps to reach for non linearity in FEM applications to reach quite fast the implicit solution. It is a modification of the Newton Raphson method, that is usable for stability analysis and non linear static simulations, in particular in the domain of limit states and for buckling and post buckling. As one can see from the graph that opposite to the Newton Raphson method the aim of the equilibrium integrations is not the load amplitude but approaches the solution on a curve. This arc represents an equilibrium curve derived from the DOF values and a loading factor. For applications with large changes the radius of the acr (load factor) should be small enough. The arclength method does not work well for abrupt, unsteady non linearity like contact or elastic ideally plastic material. Variants are the Riks method. 39
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The most popular method for integrating the plasticity equations is radial return mapping. Initially developed for J2 plasticity, it was generalized to arbitrary convex yield surfaces and yield stress functions. The principle is simple: First the stress is updated assuming that the response is elastic, then if it is outside the yield surface, the stress is projected on to the closest point of the yield surface. If the material is perfectly plastic, the yield surface is constant, but if h > 0, the yield surface expands during the plastic flow, and the stress is projected on the expanded yield surface. The data available at the beginning of the radial return update is the previous stress, equivalent plastic strain, and increment in the strain. By the end of the radial return procedure, the stress and equivalent plastic strain have been updated. Main drawback is the need for computing gradients of the flwo rule and hardening laws, what can be laborious for complicated plasticity models. 53
Another plastic return mapping algorithm is the Cutting plane algorithm proposed by Ortiz and Simo 1985. It avoids the need for computing the gradients of the flow rule and hardening function. The direction of the return mapping is always the steepest descent and a sequence of straight segments. Opposite to the radial return mapping, it is an explicit procedure. It converges very fast. 54
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