Selection of an Appropriate Solver

There are three main procedures for solving general structural problems in Abaqus: static and dynamic implicit solutions, and explicit dynamic solution. Other procedures exist for frequency analysis, and thermal, fluid, electrical analyses, etc. One of these procedures may be selected for any given analysis step; a ‘step’ can defined as a part of a simulation history which controls analysis procedure, loading and output options [144]. An analysis may consist of a number of steps, which may require the use of different procedures; however, explicit procedures cannot be used with ‘Standard’ (i.e. implicit) procedures within the same analysis [118]. A given solver may be more appropriate for certain problems than the others.

To aid the following discussion, two more definitions will be useful [144]: an ‘increment’ is taken to be a fraction of the total load to be applied within a step. The break-down of the total load in this manner is necessary when the structural response is non-linear. In Abaqus/Explicit, this increment is temporal, but has the same effect of splitting the applied load into fractions of the whole. An ‘iteration’ applies to implicit analyses only. This is where the solver attempts to converge on an equilibrium solution for a given increment, typically via Newton’s Method. The solver will attempt a number of iterations in an effort to reach an equilibrium condition, and, depending on the severity of the non-linear response, may require a number of reductions in the size of the increment in order to do so.

The implicit solver uses an iterative approach to solve a stiffness matrix-based problem of the general form [P] – [K][u] = 0, where [P] is the applied load vector, [K] is the global stiffness matrix and [u] is the nodal displacement vector, and the unknown that the implicit procedure will attempt to solve. The solution is dependent on the relationship between nodes, due to the matrix [K]. The implicit, dynamic solution differs in that it uses direct integration in order to better solve transient and non-linear problems, but the basis of solving a stiffness matrix-based problem via an iterative approach remains the same [144,145]. The explicit solver, on the other hand, directly integrates the kinematic behaviour of each individual node via solution of the nodal equations of motion, with the state of the node at each increment being based on the results from the previous increment [144,145]. Stated another way, an implicit procedure derives the state of the model at time t + Δt based on information at time t + Δt, whereas an explicit solver derives the state of the model at time t + Δt based on information at time t [146].

54 The explicit dynamic analysis procedure is implemented via integration of the equations of motion for a body (in this case, an individual element), used alongside a lumped element mass matrix. Thus, the equations been solved at each increment are as follows:

𝑢̇(𝑖+12) increment number i. The principle is very simple, finding the velocity and displacement of the body based on its previous state and the addition of an acceleration or velocity a half-step before the current increment. The remaining unknown in this system of equations is the acceleration of the body, defined as:

𝑢̈_(𝑖)^𝑁 = (𝑀^𝑁𝑁)⁻¹(𝑃_(𝑖)^𝑁 − 𝐼_(𝑖)^𝑁 ) (22)

where M^NJ is the matrix mass, P^J is the applied load vector and I^J is the internal force vector.

The internal force vector is assembled using the stiffness contributions of each individual element. This links the material and structural properties to the solution without needing a global stiffness matrix of the kind required by the implicit solution. The reader will immediately recognise the above equation as a rearrangement of Newton’s Second Law, F=ma, thus underlining the computational simplicity of the explicit dynamic procedure.

The implicit solver, depending on the precise settings, can be used for both static and dynamic problems, provided the energy-dissipation in the latter is minimal [118]. The explicit solver, although a dynamic approach - indeed, its original purpose was to study high-speed transient events such as impact events – is also suitable for non-linear quasi-static problems [144], provided due precautions are taken to ensure that the structural response is indeed quasi-static. These validation steps will be discussed later. Examples abound in the literature where the explicit solver is used in this manner, typically in cases where there is complex contact and/or sliding non-linearity such as that found in metal-forming problems [147,148], or considerable material non-linearity, such as that caused by material softening and failure [146,148-151]. It is material non-linearity due to damage and failure that is of concern in the current problem. The use of Abaqus/Explicit for quasi-static transverse indentation of composite sandwich panels has been successfully demonstrated by Czabaj et al [33] and Foo et al [52].

55 The implicit solver is unconditionally stable with respect to the size of the time increment [144,147,152], making it the far more efficient choice for linear and mildly non-linear quasi-static problems, since there is no upper limit on the size of the increment. For example, where one is only concerned with the elastic behaviour of a simple structure, it is perfectly plausible for the implicit solver to solve the problem in a single increment. Additionally, the implicit solver can potentially provide more accurate results [152]. The key limitation of the implicit solver is a lack of robustness, which causes convergence problems in highly non-linear problems, particularly complex contact simulations, or where there is material softening due to damage. Indeed, this can cause, in the extreme, an analysis to abort as the solver fails to find a convergent solution. This weakness is noted in a number of papers, and it is well-known that the use of an explicit solver can overcome this difficulty [56,145-147,152,153].

To explain this deficiency in more depth, within the context of the current research, recall that the material properties in the damaged region are degraded during the damage evolution phase. Once this degradation reaches its maximum, the stiffness matrix becomes singular for the elements in that region, causing the implicit solver to abort the analysis when it attempts and inevitably fails to invert the matrix. Convergence failure can be delayed via the use of additional elements through the thickness of the plate, as it would allow the material to fail in a more progressive (and indeed more realistic) manner; however, this issue is inherent within the implicit solver and cannot be eliminated: the analysis would still abort once a completely damaged region exists in each layer, even though the structure may not have actually failed.

The key disadvantage of the explicit solver is its conditional stability [144,147,152]. This creates a limit on the maximum size of a time increment, defined in terms of the highest frequency of the system, and estimated via the following simple relationship [144]:

∆𝑡𝑚𝑚𝑚 = 𝑙𝑒𝑒

𝑣𝑑

(23)

Where lel is the shortest element length, and vd is the dilatational wave speed of the material, given by:

𝑣_𝑑 = �𝐸 𝜌 (24)

Where E and ρ are respectively the stiffness and density of the material. Typically, this maximum stable time increment will be very small – for composite materials with a properly refined mesh, this can be as small as 1e^-9 s. The obvious consequence of this is that analyses with a long time step, such as quasi-static problems where a long duration is necessary to ensure a suitably low loading rate, become hugely demanding of computational resources as many thousands or millions of increments become necessary to solve the problem. As a result,

56 it is usually necessary to manipulate the solution via artificial acceleration of the loading and/or by artificially increasing the density (‘mass scaling’) [144,147] to increase the stability limit and thus improve computational efficiency. Both of these approaches change the physics of the problem being studied by increasing inertial effects within the solution, and thus need to be used with care, and checks must be performed to ensure the validity of the results is not compromised. Additionally, accelerated loading cannot be used with rate-dependant materials [118]. The stable time increment can of course be increased by using larger elements, but care must be taken to avoid an overly coarse mesh, which could also cause inaccurate results, particularly if used with cohesive elements.

If the problem is artificially altered via accelerated loading or mass scaling for computational efficiency, the solution must be checked to confirm that the results can be considered true for a quasi-static problem. The usual method is to compare the kinetic energy of the system with the internal energy of the system at the end of the step. Ideally, the kinetic energy would be zero or close to zero for a truly quasi-static case. The typical rule of thumb suggests that if the kinetic energy is less than 5% of the internal energy, the solution can be considered to be quasi-static, as inertial effects are negligible at this energy level [144,146,147]. An additional guarantee of a quasi-static solution is to compare the loading rate to the lowest natural period of the structure [144,147]. By finding the lowest natural frequency of the structure, one can find the period for this mode by taking the reciprocal of that frequency. A quasi-static deformation will occur over a much longer time period than the natural period, so, providing that the loading rate is much lower than this (at least 10 times slower is recommended) then the solution can be considered quasi-static. Such manipulations of the problem to reduce the solution runtime are generally not required for truly dynamic events, since these transient events are necessarily of very short duration, and thus can be modelled in true time without difficulty.