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The purpose of this study is to perform a detailed review on the numerical modeling of multiphase and multicomponent flows in microfluidic system using phase-field method. The phase-field method is of emerging importance in numerical computation of transport phenomena involving multiple phases and/or components. This method is not only used to model interfacial phenomena typical to multiphase flows encountered in engineering and nature but also turns out to be a promising tool in modeling the dynamics of complex fluid-fluid interfaces encountered in physiological systems such as dynamics of vesicles and red blood cells). Intrinsically, a priori unknown topological evolution of interfaces offers to be the most concerning challenge toward accurate modeling of moving boundary problems. However, the numerical difficulties can be tackled simultaneously with numerical convenience and thermodynamic rigor in the paradigm of the phase field method.

The phase-field method replaces the macroscopically sharp interfaces separating the fluids by a diffuse transition layer where the interfacial forces are smoothly distributed. As against the moving mesh methods (Lagrangian) for the explicit tracking of interfaces, the phase-field method implicitly captures the same through the evolution of a phase-field function (Eulerian). In contrast to the deployment of an artificially smoothing function for the interface as used in the volume of a fluid or level set method, however, the phase-field method uses mixing free energy for describing the interface. This needs the consideration of an additional equation for an order parameter. The dynamic evolution of the system (equation for order parameter) can be described by Allen–Cahn or Cahn–Hilliard formulation, which couples with the Navier–Stokes equation with the aid of a forcing function that depends on the chemical potential and the gradient of the order parameter.

In this review, first, the authors discuss the broad motivation and the fundamental theoretical foundation associated with phase-field modeling from the perspective of computational microfluidics. They subsequently pinpoint the outstanding numerical challenges, including estimations of the model-free parameters. They outline some numerical examples, including electrohydrodynamic flows, to demonstrate the efficacy of the method. Finally, they pinpoint various emerging issues and futuristic perspectives connecting the phase-field method and computational microfluidics.

This paper gives unique perspectives to future directions of research on this topic.

The purpose of this paper is to develop an efficient numerical method to study the complex crack initiation and propagation in linear elastic multiphase composites.

A phase field method is developed to study the complex fracture behavior in multiphase composites. A damage threshold is introduced for referring crack initiation in the proposed method. The damage threshold is assigned as a material property so that different composite components possess different thresholds. In this manner, smooth transition from crack initiation to propagation is revealed.

The proposed method is used to investigate complex crack evolution in mesoscale cementitious composite, which consists of aggregates, matrix and void pores. From a mesoscale point of view, it is found that cracks prefer to evolve within the matrix phase. As a crack encounters an aggregate, it tends to bypass the aggregate and evolve along the interface. Cracks tend to avoid to penetrate through aggregates. Also, cracks tend to be attracted by void pores. From a mesoscale point of view, it is revealed that the elastic modulus and strength of concrete models are closely related to porosity.

A criterion with a damage threshold is introduced to the proposed method. The criterions with and without a damage threshold are compared with each other in details. The proposed method is proven to be a useful tool to study mechanical behavior and crack evolution of brittle multiphase composites.

The purpose of this study is to develop a new numerical algorithm to simulate the phase-field model.

First, the derivative of the temporal direction is discretized by a second-order linearized finite difference scheme where it conserves the energy stability of the mathematical model. Then, the isogeometric collocation (IGC) method is used to approximate the derivative of spacial direction. The IGC procedure can be applied on irregular physical domains. The IGC method is constructed based upon the nonuniform rational B-splines (NURBS). Each curve and surface can be approximated by the NURBS. Also, a map will be defined to project the physical domain to a simple computational domain. In this procedure, the partial derivatives will be transformed to the new domain by the Jacobian and Hessian matrices. According to the mentioned procedure, the first- and second-order differential matrices are built. Furthermore, the pseudo-spectral algorithm is used to derive the first- and second-order nodal differential matrices. In the end, the Greville Abscissae points are used to the collocation method.

In the numerical experiments, the efficiency and accuracy of the proposed method are assessed through two examples, demonstrating its performance on both rectangular and nonrectangular domains.

This research work introduces the IGC method as a simulation technique for the phase-field crystal model.

This paper develops a neural network surrogate model based on a discrete lattice approach to investigate the influence of complex microstructures on the emergent behavior of biological networks.

The adaptability of network-forming organisms, such as, slime molds, relies on fluid-to-solid state transitions and dynamic behaviors at the level of the discrete microstructure, which continuum modeling methods struggle to capture effectively. To address this challenge, we present an optimized approach that combines lattice spring modeling with machine learning to capture dynamic behavior and develop nonlinear constitutive relationships.

This integrated approach allows us to predict the dynamic response of biological materials with heterogeneous microstructures, overcoming the limitations of conventional trial-and-error lattice design. The study investigates the microstructural behavior of biological materials using a neural network-based surrogate model. The results indicate that our surrogate model is effective in capturing the behavior of discrete lattice microstructures in biological materials.

The combination of numerical simulations and machine learning endows simulations of the slime mold Physarum polycephalum with a more accurate description of its emergent behavior and offers a pathway for the development of more effective lattice structures across a wide range of applications.

The novelty of this research lies in integrating lattice spring modeling and machine learning to explore the dynamic behavior of biological materials. This combined approach surpasses conventional methods, providing a more holistic and accurate representation of emergent behaviors in organisms.

The purpose of this paper is to analyze the repair of a straight and angular crack in the structure using a piezoelectric material under thermo-mechanical loading by the extended finite element method (XFEM) approach. This provides a general and simple solution for the modeling of crack in the structure to analyze the repair.

The extended finite element method is used to model crack geometry. The crack surface is modeled by Heaviside enrichment function while the crack front is modeled by branch enrichment functions.

The effectiveness of the repair is measured in terms of stress intensity factor and J-integral. The critical voltage at which patch repair is most effective is evaluated and presented. Optimal patch shape, location of patch, adhesive thickness and adhesive modulus are obtained for effective repair under thermo-mechanical loading environment.

The presented numerical modeling and simulation by the XFEM approach are of great benefit to analyze crack repair in two-dimensional and three-dimensional structures using piezoelectric patch material under thermo-mechanical loading.

The purpose of this paper is to present an optimization method for flux barrier designs in interior permanent magnet (IPM) synchronous motors that aims to produce an advantageous sinusoidal flux density distribution in the air-gap.

The optimization is based on the phase field method using an Allen-Cahn equation. This approach is a numerical technique for tracking diffuse interfaces like the level set method based on the Hamilton-Jacobi equation.

The optimization results of IPM motor designs are highly dependent on the initial flux barrier shapes. The authors solve the optimization problem using two different initial shapes, and the optimized models show considerable reductions in torque pulsation and the higher harmonics of back-electromotive force.

This paper presents the optimization method based on the phase field for the design of rotor flux barriers, and proposes a novel interpolation scheme of the magnetic reluctivity.

This paper aims to present a nonlocal gradient plasticity damage model to demonstrate the crack pattern of a body, in an elastic and plastic state, in terms of damage law. The main objective of this paper is to reconsider the nonlocal theory by including the material in-homogeneity caused by damage and plasticity. The nonlocal nature of the strain field provides a regularization to overcome the analytical and computational problems induced by softening constitutive laws. Such an approach requires C1 continuous approximation. This is achieved by using an isogeometric approximation (IGA). Numerical examples in one and two dimensions are presented.

In this work, the authors propose a nonlocal elastic plastic damage model. The nonlocal nature of the strain field provides a regularization to overcome the analytical and computational problems induced by softening constitutive laws. An additive decomposition of strains in to elastic and inelastic or plastic part is considered. To obtain stable damage, a higher gradient order is considered for an integral equation, which is obtained by the Taylor series expansion of the local inelastic strain around the point under consideration. The higher-order continuity of nonuniform rational B-splines (NURBS) functions used in isogeometric analysis are adopted here to implement in a numerical scheme. To demonstrate the validity of the proposed model, numerical examples in one and two dimensions are presented.

The proposed nonlocal elastic plastic damage model is able to predict the damage in an accurate manner. The numerical results are mesh independent. The nonlocal terms add a regularization to the model especially for strain softening type of materials. The consideration of nonlocality in inelastic strains is more meaningful to the physics of damage. The use of IGA framework and NURBS basis functions add to the nonlocal nature in approximations of the field variables.

The method can be extended to 3D. The model does not consider the effect of temperature and the dissipation of energy due to temperature. The method needs to be implemented for more real practical problems and compare with experimental work. This is an ongoing work.

The nonlocal models are suitable for predicting damage in quasi brittle materials. The use of elastic plastic theories allows to capture the inelastic deformations more accurately.

The nonlocal models are suitable for predicting damage in quasi brittle materials. The use of elastic plastic theories allows to capture the inelastic deformations more accurately.

The present work includes the formulation and implementation of a nonlocal damage plasticity model using an isogeometric discretization, which is the novel contribution of this paper. An implicit gradient enhancement is considered to the inelastic strain. During inelastic deformations, the proposed strain tensor partitioning allows the use of a distinct potential surface and distinct failure criterion for both damage and plasticity models. The use of NURBS basis functions adds to more nonlocality in the approximation.

The purpose of this paper is to develop a coupled lattice Boltzmann model for the simulation of the freezing process in unsaturated porous media.

In the developed model, the porous structure with complexity and disorder was generated by using a stochastic growth method, and then the Shan-Chen multiphase model and enthalpy-based phase change model were coupled by introducing a freezing interface force to describe the variation of phase interface. The pore size of porous media in freezing process was considered as an influential factor to phase transition temperature, and the variation of the interfacial force formed with phase change on the interface was described.

The larger porosity (0.2 and 0.8) will enlarge the unfrozen area from 42 mm to 70 mm, and the rest space of porous medium was occupied by the solid particles. The larger specific surface area (0.168 and 0.315) has a more fluctuated volume fraction distribution.

The concept of interfacial force was first introduced in the solid–liquid phase transition to describe the freezing process of frozen soil, enabling the formulation of a distribution equation based on enthalpy to depict the changes in the water film. The increased interfacial force serves to diminish ice formation and effectively absorb air during the freezing process. A greater surface area enhances the ability to counteract liquid migration.

This study implements the fourth-order phase field method (PFM) for modeling fracture in brittle materials. The weak form of the fourth-order PFM requires C¹ basis functions for the crack evolution scalar field in a finite element framework. To address this, non-Sibsonian type shape functions that are nonpolynomial types based on distance measures, are used in the context of natural neighbor shape functions. The capability and efficiency of this method are studied for modeling cracks.

The weak form of the fourth-order PFM is derived from two governing equations for finite element modeling. C⁰ non-Sibsonian shape functions are derived using distance measures on a generalized quad element. Then these shape functions are degree elevated with Bernstein-Bezier (BB) patch to get higher-order continuity (C¹) in the shape function. The quad element is divided into several background triangular elements to apply the Gauss-quadrature rule for numerical integration. Both fourth-order and second-order PFMs are implemented in a finite element framework. The efficiency of the interpolation function is studied in terms of convergence and accuracy for capturing crack topology in the fourth-order PFM.

It is observed that fourth-order PFM has higher accuracy and convergence than second-order PFM using non-Sibsonian type interpolants. The former predicts higher failure loads and failure displacements compared to the second-order model due to the addition of higher-order terms in the energy equation. The fracture pattern is realistic when only the tensile part of the strain energy is taken for fracture evolution. The fracture pattern is also observed in the compressive region when both tensile and compressive energy for crack evolution are taken into account, which is unrealistic. Length scale has a certain specific effect on the failure load of the specimen.

Fourth-order PFM is implemented using C¹ non-Sibsonian type of shape functions. The derivation and implementation are carried out for both the second-order and fourth-order PFM. The length scale effect on both models is shown. The better accuracy and convergence rate of the fourth-order PFM over second-order PFM are studied using the current approach. The critical difference between the isotropic phase field and the hybrid phase field approach is also presented to showcase the importance of strain energy decomposition in PFM.

The phase-field method of interfacial damage is used to simulate the damage in composite structures containing the brittle orthotropic materials and their interface.

In the brittle fracture modeling, the strain tensor is decomposed into positive and negative parts characterizing tension and compression behaviors. By requiring an elastic energy preserving transformation involving the elastic stiffness tensor, these two strain parts must satisfy the orthogonality condition in the sense that the elastic stiffness tensor responds as a metric. However, most of the recent phase-field methods for brittle fracture do not verify this orthogonality condition. Additionally, to describe the damage in structures with anisotropic phases, recent studies have used multiple phase-field variables, with each preferential orientation represented by a phase-field variable to describe the bulk damage of component materials. This approach increases the complexity of simulation procedure. These disadvantages motivate the present study aimed at enhancing the simulation method.

The present study improves the phase-field method of interfacial damage by (1) incorporating the strain orthogonality condition into the phase-field method; (2) using only one phase-field variable instead of multiple phase-field variables to simulate damage in component orthotropic phases; and (3) investigating the interaction between interfacial damage and bulk damage as well as the effect of orientation tensor of preferential orientation in each orthotropic phase and the interfacial parameters on crack branching in composite structures.

Through several simulation examples, the present simulation method is proven to be accurate, effective, and helps the simulation process simpler than previous relevant methods.