In the FEA solver world, users come across multiple numerical schemes to solve the formulated stiffness matrix of the problem. The most popular ones among all are the implicit and explicit solvers. In Abaqus terminology they are called a standard solver and explicit solver respectively. Each of these schemes has its own merits and demerits and this blog post compares these two schemes based on several parameters.

For ease of understanding, I am avoiding the use of long and complicated mathematical equations in this post. 😉

** Implicit Scheme**

From an application perspective, this scheme is primarily used for static problems that do not exhibit severe discontinuities. Let’s take an example of the simplest problem: Linear static in which any physical situation can be mathematically formulated as:

[K]{x}={F}Here K is the stiffness matrix, x is the displacement vector and F is the load vector. The size of the matrix and vectors can vary depending on the dimensionality of the problem. For example, K can be a 6×6 matrix for a 3D continuum problem or a 3×3 matrix for a 2D structural problem. The composition of K is primarily governed by material properties. F primarily includes forces and moments at each node of the mesh. Now, to solve the above equation for x, matrix K should be inverted or inversed. After inversion, we get a displacement solution used to compute other variables, such as strains, stresses, and reaction forces.

[M]d^{2}{x}/dt

^{2}+[C]d{x}/dt+[K]{x}={F}

The Implicit scheme is applicable to dynamic problems as well. In the above equation, M is mass matrix, C is damping matrix and the rest are as usual. This equation is defined in real time. **Backward Euler** time integration is used to discretize this equation in which the state of a system at a given time increment depends on the state of the system at later time increment. K matrix inversion takes place in a dynamic scenario as well because the objective is still to solve for x. Abaqus standard solver uses three different approaches to solve implicit dynamic problems: quasi static, moderate dissipation or transient fidelity. Each method is recommended for specific types of non-linear dynamic behavior. For example, the quasi static method works well in problems with severe damping.

**Merits of this scheme**

- For linear problems, in which K is a constant, implicit scheme provides solution in a single increment.
- For non-linear problems, in which K is a function of x, thereby making it necessary to solve problem in multiple increments for sake of accuracy, size of each increment can be considerably large as this scheme is unconditionally stable.

Due to these reasons, implicit scheme is preferred to simulate linear/non-linear static problems that are slow or moderate in nature with respect to time.

**Demerits of this scheme** […]