We propose a novel computational framework that is capable of employing different time integration algorithms and different space discretized methods such as the Finite Element Method, particle methods, and other spatial methods on a single body sub-divided into multiple subdomains. This is in conjunction with implementing the well known Generalized Single Step Single Solve (GS4) family of algorithms which encompass the entire scope of Linear Multistep algorithms that have been developed over the past 50 years or so and are second order accurate into the Differential Algebraic Equation framework. In the current state of technology, the coupling of altogether different time integration algorithms has been limited to the same family of algorithms such as the Newmark methods and the coupling of different algorithms usually has resulted in reduced accuracy in one or more variables including the Lagrange multiplier. However, the robustness and versatility of the GS4 with its ability to accurately account for the numerical shifts in various time schemes it encompasses, overcomes such barriers and allows a wide variety of arbitrary implicit-implicit, implicit-explicit, and explicit-explicit pairing of the various time schemes while maintaining the second order accuracy in time for not only all primary variables such as displacement, velocity and acceleration but also the Lagrange multipliers used for coupling the subdomains. By selecting an appropriate spatial method and time scheme on the area with localized phenomena contrary to utilizing a single process on the entire body, the proposed work has the potential to better capture the physics of a given simulation. The method is validated by solving 2D problems for the linear second order systems with various combination of spatial methods and time schemes with great flexibility. The accuracy and efficacy of the present work have not yet been seen in the current field, and it has shown significant promise in its capabilities and effectiveness for general linear dynamics through numerical examples.

As numerical analysis is utilized in the various fields and applications, the desire for analyzing complex problems more efficiently has grown. The Finite Element Method (FEM) is one of the most widely used numerical analysis methods. While it features simplicity and versatility, it is susceptible to numerical errors such as shear locking and complexity in adopting it for large deformation problems and the like.

Concurrently, the strong form based particle methods has been gaining interest in the field of numerical simulations as the particle methods have the capability and flexibility for handling large distortion, crack propagation and free surface detecting, etc. Some of the notable strong form particle methods, Smoothed Particle Hydrodynamics (SPH) [

As problems become more complex, controlling the spatial discretization locally is a desirable feature. While an area where fast dynamics are presented is discretized with high spatial resolutions, a coarser mesh may be sufficient in the majority of the structure. In addition, the selection of spatial methods in different parts of a body can also be an effective strategy such as by utilizing the FEM for the majority of a body, and applying particle based methods only on an area where localized phenomena such as crack or fracture may occur.

One of the ways to locally control spatial discretization is dividing a body into regions of subdomains in which different methods can be applied in each subdomain. The challenge in using subdomains is handling the interface condition between the subdomains, and over the years, numerous coupling methods have been developed. Interface coupling is largely divided into two categories: overlapping and non-overlapping domain coupling. In the case of the overlapping domain decomposition, a portion of the subdomains are overlapped and the shape function similar to the framework of the FEM is used to constrain the common Degrees of Freedom (DOFs). However, the large difference in each of the subdomain’s stiffness may lead to the global stiffness matrix to be ill-conditioned, as noted by Orsini et al. in [

On the other hand, the non-overlapping technique is more straight forward where the subdomains are only connected by an edge or a surface in 2D or 3D, respectably, and are constrained by the Lagrange multiplier at the interface. At the interface, usually the nodes/particles are overlapped; however, Park et al. [

Implementing the different spatial discretization methods for localized phenomena has been a study which has been explored for many years. The coupling of the FEM with molecular dynamics via a bridging domain which enforces compatibility by utilizing the Lagrange multiplier is proposed by Xiao et al. [

There also has been high interest in coupling subdomains for transient analysis. Gravouil et al. presented the GC method in [

Similarly, Pegon extended the GC method and proposed the PM method which employs an inter-field parallel solution procedure to couple different subdomains [

In this work, the subdomain DAE framework is particularly advanced to the Generalized Single Step Single Solve (GSSSS or GS4) framework and family of algorithms developed by Tamma et al. [

A brief overview of the finite element method is reviewed in this section for better understanding of the various numerical examples and its methodology that are presented in a later section. In addition, the objective is to get clarity of the computations involved in the calculations of the FEM integrals and particle method integrals.

In the finite element method, a body of domain

Quadrilateral elements with two Gauss points in each x and y direction will be used throughout in the numerical example section.

The equilibrium equation expressed in terms of the displacement can be written as (shown for 1D as a simple illustration)

Next we introduce an arbitrary function

Then we integrate over the domain to obtain an integral form that is set to zero.

The stress term is then integrated by parts as

Introduce the constitutive equation to obtain governing equation based on displacement, and we get

It is worth noting that we removed

The resulting weak formulation in space is given as

Setting

The above equation is valid for any

A physical field can be thought of as the assignment of a physical quantity at each point in space and time. Consider a domain in Euclidean space,

The weighted residual method has been widely used in solving different kinds of ordinary/partial differential equations, in particular, using the Galerkin method, which is the fundamental theory of the finite element method. In this work, we exploit the weighted residual method directly to the Taylor series expansion, such that a generalized approach of developing particle based methods can be explored.

In mathematics, the Taylor series expansion is a representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point. The generalized Taylor series of a general physical quantity

Omitting the higher order terms from

According to the standard weighted residual method, we introduce a weight function, ^{1}

In this work, the weight function used in the weighted residual method is defined as

It is worth noting that the domain is the cut-off influence domain of the position

By treating

Hence, the gradient of a vector can be obtained as

The proposed gradient operator has the ability to recover the ^{′}^{′}

Let

Discretizing into particles, and applying

Thus,

Gradient Operator:

Introducing the weighting function

It is worth noting that for a general particle within a uniformly distributed system, the weighted average matrix

The discretized formulations is listed as following

In order to obtain the Laplacian relationship between the center point

However, the rank of matrix

Therefore, an alternative gradient of a vector is given by taking advantage of the average property,

Consequently, the Laplacian can be obtained via combining Gauss’s Theorem and the gradient formulations.

Therefore, the Laplacian of

Hence the following Laplacian formulations can be obtained as

The discretized formulations is listed as following:

We will explore how the stiffness matrix is formed in 2D using the GPS method. We start with the same governing equation and constitutive law,

The first step to get the stiffness matrix is to describe ∇

Thus,

Next, we need to convert the 2 by 2 matrix on the left hand side of

We can also get the engineering strain as follows:

Stress

It is worth noting that [

Next, we will show how to construct the divergence operator into a matrix form similar to how the gradient operator matrix was formed as shown above.

Thus, we can get the stiffness matrix for the particle 1, by combining

This process is repeated for each particle in the system. After generating

Differential Algebraic Equations (DAE) are differential equations with algebraic constraints. Consider the equation of motion of a body that is subject to constraints

Now, consider a domain of interest

The matrix

Let

Now applying the GS4 family of algorithms, the equation of motion of each subdomain can be written as following:

It is worth noting that the Lagrange multipliers are approximated in the system time level rather than in the subdomain time level. This factor suggests that the solution must be in the system time level even though the displacement, velocity, and acceleration are approximated in the subdomain time step.

The updates of each of the unknown variables with the GS4 framework of algorithms are

In linear dynamical systems,

A set of all kinematic unknowns for

The constraint equation at the velocity level, i.e.,

The equation of motion for the entire system for a system time step level can be represented in a simple matrix form as following:

The computation flowchart for the GS4-II DAE framework for the subdomain method employing different spatial methods, algorithms, and time steps is shown below in

_{dom}

Calculate

Calculate _{i}_{i}

Calculate Ĝ_{i} using

Assemble _{i} using

Assemble _{i} using

Assemble

Assemble

Transpose

_{dom}

Calculate _{i}

Assemble

Assemble ^_{glob} and _{glob} from

Solve (_{glob}_{n+1} = ^_{glob})

The second order time accuracy of all the primary variables and the Lagrange multipliers can be achieved for interfacing different subdomains (this is in general not plausible with traditional methods) if

The GS4-II algorithms are written in the form of U0(

With the

Case | Subdomain 1 | Subdomain 2 |
---|---|---|

1 | U0(1, 1, 0) (Newmark) | U0(0, 0, 0) (WBZ) |

2 | U0(1, 1, 0) (Newmark) | U0(0, 1, 0) (U0V0 optimal) |

3 | U0(0.5, 1, 0.5) (U0V0 optimal) | U0(0.5, 0.5, 0.5) (Three parameter optimal) |

4 | V0(1, 1, 0) (MPR-MPA) | V0(1, 1, 1) (MPR-EPA) |

5 | V0(0.7, 0.7, 0.7) | V0(0.7, 0.7, 0.21) |

For the U0 family of GS4-II algorithms, the

The bar is compressed from the right side with p, the distributed load, of 10,000 N/m. This material has 72 GPa for Young’s modulus, 0.3 for Poisson’s ratio, and

To further present evidence of coupling various time scheme algorithms, five different cases listed in

Lastly, the subcycling cases are also evaluated in

A beam with the same geometry as the one in the wave propagation problem is used to illustrate the application of the multi-domain method in a plane stress dynamic problem. A cantilever beam of 10 m by 1 m with 1 m thickness is fixed on the left side and is experiencing downward traction, p, of 100,000

The system

Similar to the previous problem, a different combination of methods are compared in

Next, a body divided into three domains is examined next for the extension of the subdomain coupling concept. The same vibration problem above is evaluated again but this time, the beam is divided into three subdomains where

The bar is fixed at the left end and pulled down from the right end with a distributed load of 210 N/m. This material has 70 GPa for Young’s modulus, 0.3 for Poisson’s ratio, and

Case | Subdomain 1 | Subdomain 2 | Subdomain 3 |
---|---|---|---|

1 | FEM U0(1, 1, 0) | Peridynamics U0(0, 0, 0) | MPS U0(1, 1, 0) |

2 | FEM U0(0.8, 0.5, 0.1) | Peridynamics U0(0.6, 0.3, 0.1) | MPS U0(1, 0.7, 0.1) |

3 | FEM V0(0.8, 0.5, 0) | Peridynamics V0(0.8, 0.5, 0.2) | MPS V0(0.8, 0.5, 0.4) |

4 | FEM U0(1, 1, 0.7) | Peridynamics U0(1, 1, 0.1) | MPS U0(1, 1, 0.6) |

The time history of the y-direction displacement, velocity and acceleration of nodes/particles in each subdomain (points A, B, and C in

In this paper, we have proposed a novel implementation of the Generalized Single Step Single Solve time integration framework for ODE’s into the DAE framework. This proposed method, unlike traditional approaches in current technology with LMS methods, allows the coupling of different numerical analysis methods such as FEM and particle methods to be used on a single body while ensuring the accuracy of the result with the use of a wide variety of time integration algorithms within the GS4 framework. For example, one cannot simply or haphazardly couple the Midpoint rule with the Newmark method; or couple a multitude of schemes such as the Newmark, Midpoint rule, HHT, WBZ methods, etc., arbitrarily in different subdomains in a single analysis without loss and reduced accuracy in the convergence rate of one or more variables. However the GS4 framework circumvents various deficiencies, preserves accuracy, and permits a much broader design space for coupling algorithms in a single analysis and provides much desired robust features for applications to large scale industrial real world problems as well. It has the potential to increase the accuracy of the physics by selecting an appropriate method for only the area with a localized phenomena rather than utilizing only a single method that may not be suitable for certain applications on the whole body. The method is verified by applying it to 2D simple bar and beam problems. The results from various combination of methods and time scheme algorithms match closely with the reference result. With the basis of the proposed computational methodology established, the extension of the method for the future studies includes computational efficiency studies, extension to nonlinearity, and implementation of reduced order modeling. This work provides generality and versatility of the computational framework incorporating a wide variety of subdomain based spatial and time integration algorithms in a single analysis with great accuracy.