# Macro-Micro-Coupled Simulations of Dilute Viscoelastic Fluids
## 1 Introduction  
Viscoelastic fluids are materials that exhibit complex behaviors due to their unique composition and structure. Uses of these materials can be found in many industrial, biomedical, and basic science applications [1; 2; 3; 4; 5; 6]. Common examples include liquid crystals, polymers, and colloids. In practice, modeling the behavior of viscoelastic materials requires significant computational power, which clashes with the need for fast simulations. An increasing number of research studies focus on the mathematical modeling of viscoelastic fluids at different scales. To allow such modeling efforts to be useful in practice, there is a critical need for more efficient computational methods. In recent years, GPU computing has emerged as a valuable tool to overcome these challenges. Here, we describe numerical algorithms aimed at using GPU computing in viscoelastic flow simulations. GPU computing refers to the use of graphics processing units (GPUs) to perform numerical computations. GPU computing has become an important tool for the study of complex fluids due to its ability to handle large amounts of data and perform computations in parallel. Because GPUs can handle massive computational demands, they can provide results in a fraction of the time of traditional CPU-based methods. As computing power continues to increase, GPU-based simulations can become increasingly important for understanding the behavior of complex materials and complex flows. GPUs have been previously used in the modeling of polymeric systems. [PERSON] et al. [7] performed numerical simulations to find the equilibrium properties of polymer rings. [PERSON] et al. [8] used GPUs to accelerate computations of smoothed particle hydrodynamics (SPH) for viscoelastic materials. [PERSON] et al. [9] extended their Lagrangian-Eulerian method for viscoelastic flow simulations to GPU computing. [PERSON] et al. [10] simulated phase separation in polymer blends and polymer solutions using a GPU-based viscoelastic model. [PERSON] et al. [11] introduced a lattice-Boltzmann-based finite-volume formulation, where the macroscopicequations were solved by a finite-volume method and the microscopic equation was solved by a lattice-Boltzmann method. These authors implemented the solution to the microscopic equation in GPUs. [PERSON] et al. [12] used GPUs to accelerate their three-dimensional numerical simulations of viscoelastic fluids for both the Oldroyd-B and FENE-P models. [PERSON] and [PERSON] [13] simulated ring polymer melts via GPU computing using an elastic lattice polymer model. [PERSON] et al. [14] investigated high-molecular-weight entangled polymer melts using HOCMD-blue, a GPU accelerated simulation software. In this paper, we use GPU computing to couple dynamics at the macroscale with those at the mesoscale, described by Langevin-type equations. From a modeling perspective, it is well-known that the stress, \(\mathbf{\tau}\), of any Newtonian fluid is directly proportional to the imposed shear rate, \(\mathbf{\tau}=\eta\dot{\gamma}\), where the proportionality constant \(\eta\) is the viscosity of the material. Differences between materials will only affect the value of \(\eta\) but not the functional form of the constitutive equation. Mathematically speaking, this means that for Newtonian fluids, there is a single, general constitutive equation capable of describing different types of Newtonian materials. In principle, this means that the same numerical algorithms can be used for any material by only changing the value of the viscosity constant. In contrast, there is not a single all-encompassing constitutive equation for viscoelastic materials. This is because, in this case, the evolution of the stress as a function of the applied strain can vary from material to material or even for the same material under different flow conditions. Accordingly, the formulation of constitutive equations describing different types of viscoelastic materials is a prolific area of research [15].