#### Table of Contents

Abaqus is a popular finite element analysis (FEA) software used by engineers and researchers to simulate various physical phenomena. It offers a wide range of solvers to handle different types of problems, including linear and nonlinear statics, dynamics, fluids, electromagnetics, acoustics, and thermal analyses. However, each solver has its own strengths and weaknesses, and choosing the right solver for a specific problem can significantly impact the accuracy, efficiency, and cost of the simulation. In this content, we have a comparison of different solvers available in Abaqus:

## 1. Standard Solver

The most commonly used solver in Abaqus is the standard solver, which suits linear and nonlinear static problems. It uses an incremental-iterative approach to solve the equations of motion, which makes it efficient for problems with small to moderate deformations and material nonlinearities. The standard solver can handle problems with various boundary conditions, contact, and geometries, and can output stress and strain fields, displacements, and reaction forces. However, it may not be suitable for problems with large deformations and dynamic loading, as it may require numerous iterations to converge.

The standard solver in ABAQUS uses an implicit integration scheme to solve the equations of motion for a given problem. The equations of motion may include static equilibrium equations, dynamic equations, or a combination of both.

### Formulation:

The standard solver in ABAQUS formulates the equations of motion in a matrix equation of the form:

[K]{x} = {F}

Where [K] is the stiffness matrix, {x} is the displacement vector, and {F} is the nodal force vector. The stiffness matrix [K] is assembled from the element stiffness matrices and boundary conditions, while the nodal force vector {F} is obtained from the external loads and boundary conditions.

The solution procedure updates the displacement vector {x} at each time step using a time-stepping algorithm. The algorithm initiates by making an initial guess for the displacement vector, and then iteratively solves for the displacement vector using a Newton-Raphson scheme until it reaches convergence.

The Newton-Raphson scheme involves the following steps:

- Compute the residual vector {R} = {F} – [K]{x}.
- Compute the tangent stiffness matrix [Kt] = d{R}/d{x}.
- Solve the linear system [Kt]{Δx} = {R} for the displacement increment {Δx}.
- Update the displacement vector {x} = {x} + {Δx}.
- Check for convergence criteria, such as the norm of the residual vector or the relative change in the displacement vector. If convergence is not achieved, repeat steps 1-4 with a new guess for the displacement vector.

ABAQUS uses an implicit integration scheme that is unconditionally stable, which means that one can choose the time step size independently of the size and stiffness of the problem. However, smaller time steps may be required for problems with high-frequency content or large deformations to ensure accurate results.

## 2. Explicit Solver:

The explicit solver is suitable for solving highly dynamic problems such as impact, crash, and high-speed machining. It is a time-based solver that solves the equations of motion for each time step, which makes it computationally expensive but accurate for problems with large deformations, material nonlinearities, and contact. The explicit solver can handle problems with complex geometries and boundary conditions, and can output detailed information on the deformation, strain rate, and stress state of the model. However, it may require a large number of time steps and high computational resources to produce accurate results.

Engineers and researchers use the explicit solver in ABAQUS to solve problems that involve short duration, high energy events such as crash or impact simulations. Unlike the standard solver, the explicit solver uses an explicit time integration scheme to solve the equations of motion for a given problem. The formulation of the explicit solver involves the following steps:

- The discretization process involves discretizing the domain of the problem into finite elements and defining the nodal coordinates and element connectivity.
- Element formulations: The element formulations define the equations that relate the nodal displacements, velocities, and accelerations within each element. ABAQUS uses various element types, such as beam, shell, and solid elements, each with its own set of equations.
- Combining the element equations, which relate the applied loads to the nodal velocities and accelerations, assembles the global mass matrix and damping matrix.
- The engineers or researchers apply the boundary conditions, such as fixed displacements and applied loads, to the global mass matrix and damping matrix. Initial conditions, such as the initial velocity and acceleration of the structure, are also specified.
- Engineers or researchers integrate the equations of motion over time using an explicit time integration scheme, such as the central difference method or the Newmark method. The explicit time integration scheme computes the nodal displacements, velocities, and accelerations at each time step.
- After the simulation, engineers or researchers analyze the behavior of the studied structure or system by post-processing the results, such as stresses, strains, and displacements.

Unlike the standard solver, the explicit solver does not require the solution of a linear system of equations at each time step. As a result, it is computationally more efficient for problems that involve short time scales and large deformations. However, the explicit solver may require smaller time steps than the standard solver to ensure numerical stability. Moreover, it may not be suitable for problems that involve long-duration events or quasi-static behavior.

### Formulation:

One can write the equation of motion for a finite element in the explicit solver as:

[M]{a} + [C]{v} + [K]{u} = {f}

Where [M], [C], and [K] are the mass, damping, and stiffness matrices for the element, respectively. {u}, {v}, and {a} are the nodal displacement, velocity, and acceleration vectors, respectively, and {f} is the nodal force vector. The nodal force vector {f} includes contributions from external loads and boundary conditions.

The explicit time integration scheme used in the explicit solver in ABAQUS is a second-order accurate central difference method, which has the following form:

{u}(t+Δt) = {u}(t) + Δt{v}(t) + (Δt^2/2){a}(t)

{v}(t+Δt) = {v}(t) + (Δt/2)({a}(t) + {a}(t+Δt))

The nodal displacement and velocity vectors at time t+Δt are {u}(t+Δt) and {v}(t+Δt), respectively, where Δt represents the time step size. One can obtain the nodal acceleration vectors {a}(t) and {a}(t+Δt), respectively at times t and t+Δt, by solving the equation of motion using the nodal displacement and velocity vectors.

Engineers or researchers can obtain the nodal acceleration vector by rearranging the equation of motion as:

{a} = [M]^-1({f} – [C]{v} – [K]{u})

where [M]^-1 is the inverse of the mass matrix, which can be pre-computed and stored. Updating the nodal displacement and velocity vectors is possible at each time step using the central difference method by engineers or researchers.

The explicit solver in ABAQUS also includes various numerical techniques to improve the stability and accuracy of the solution, such as:

- Engineers or researchers employ this technique to prevent hourglassing, a phenomenon in which elements undergo non-physical deformation due to a lack of internal resistance to distortion. Hourglass control methods introduce additional forces or constraints to the element to prevent such distortions.
- Contact algorithms: ABAQUS includes various contact algorithms to accurately simulate contact and penetration between bodies in contact.
- Adaptive time stepping: This technique adjusts the time step size dynamically to ensure numerical stability and accuracy.

ABAQUS explicit solver uses mathematical formulations and numerical techniques for accurate and efficient dynamic problem-solving.

## 3. CFD Solver:

The CFD solver is used for solving fluid dynamics problems such as fluid-structure interaction, flow around objects, and turbulence. It can handle problems involving steady-state and transient flows, laminar and turbulent flows, and multiphase flows. CFD solver uses finite volume method to discretize governing equations and output parameters like velocity, pressure, temperature, and turbulence intensity. However, it may require high computational resources and specialized knowledge in fluid mechanics to set up and validate the model.

Engineers/researchers use ABAQUS CFD solver to simulate fluid flows and heat transfer in various industries and applications. The formulation of the CFD solver in ABAQUS involves the following steps:

- Discretization of the domain: Engineers/researchers define nodal coordinates and element connectivity by discretizing problem domains into finite volumes/elements.
- Governing equations: Engineers or researchers need to define governing equations (e.g. Navier-Stokes) to describe fluid flow/heat transfer physics. ABAQUS can solve both laminar and turbulent flows using various turbulence models.
- Boundary conditions: The boundary conditions, such as velocity, pressure, and temperature, are specified at the inlet, outlet, and other boundaries of the domain.
- Numerical methods: Numerical methods like finite volume or element methods discretize governing equations. ABAQUS uses a stabilized finite element method for solving the Navier-Stokes equations.
- Solution procedure: The equations are solved using an iterative algorithm, such as the SIMPLE algorithm or the PISO algorithm, to obtain the velocity and pressure fields.
- Post-processing: Post-processing involves analyzing fluid flow or heat transfer results, such as velocity, pressure, and temperature fields.

CFD solver in ABAQUS can simulate a range of fluid flows, including incompressible and compressible, steady-state and transient, and single and multiphase flows.. It also includes a wide range of physical models and boundary conditions to accurately simulate real-world scenarios.

ABAQUS CFD solver uses numerical algorithms and mathematical formulations to solve complex fluid flow and heat transfer problems efficiently.

## 4. Electromagnetic Solver:

Electromagnetic problems, e.g. interference, compatibility, and antenna design, can be solved using the electromagnetic solver. The solver utilizes the finite element method to discretize governing equations and can effectively handle problems with varying boundary conditions and materials. The electromagnetic solver can output various parameters such as electric and magnetic field intensities, current density, and power dissipation. However, it may require specialized knowledge in electromagnetics and high computational resources to set up and validate the model.

## 5. Acoustic Solver:

Engineers or researchers solve acoustic problems like noise prediction, sound radiation, and sound absorption with the acoustic solver. It uses finite element method and can handle problems with different boundary conditions and materials. The acoustic solver can output various parameters such as sound pressure level, sound power, and sound intensity. However, it may require specialized knowledge in acoustics and high computational resources to set up and validate the model.

## 6. Thermal Solver:

The thermal solver is used for solving thermal problems such as heat transfer, thermal stress analysis, and thermal-mechanical coupling. It employs finite element method and can handle problems with different boundary conditions, materials, and geometries. The thermal solver can output various parameters such as temperature, heat flux, and thermal stress. However, it may require high computational resources and specialized knowledge in heat transfer to set up and validate the model.

## 7. Structural Dynamics Solver:

Structural dynamics solver for earthquake, vibration, and fatigue analysis involving dynamic loading. It uses modal analysis to solve motion equations and can handle problems with different materials and boundary conditions. The structural dynamics solver can output various parameters such as natural frequencies, mode shapes, and damping ratios. However, it may require high computational resources and specialized knowledge in structural dynamics to set up and validate the model.

## Conlusion

In general, Solver choice in Abaqus depends on problem nature, accuracy, resources, and user expertise. Abaqus has standard solver for linear/nonlinear static problems; explicit solver for dynamic/large deformation problems. The CFD solver is well-suited for fluid dynamics problems, while the electromagnetic solver is ideal for electromagnetic problems with different materials and geometries. Acoustic solver for acoustic problems with various boundary conditions and sources; thermal solver for thermal problems. The structural dynamics solver may be suitable for problems with dynamic loading and structural responses.

In summary, Abaqus offers a range of solvers for different types of problems, each with its own strengths and weaknesses. The solver choice impacts accuracy, efficiency, and cost, requiring careful consideration of problem characteristics and resources.