A Review of Numerical Techniques for Frictional Contact Analysis

Abstract

This review analyzes numerical techniques for frictional contact problems, highlighting their strengths and limitations in addressing inherent nonlinearities and computational demands. Finite element methods (FEM), while dominant due to versatility, often require computationally expensive iterative solutions. Alternative methods, like boundary element methods (BEM) and meshless methods, offer potential advantages but require further exploration for broader applicability. The choice of contact algorithm significantly impacts accuracy and efficiency; penalty methods, though computationally efficient, can lack accuracy at high friction coefficients; whereas, Lagrange multiplier methods, while more accurate, are computationally more demanding. The selection of an appropriate friction constitutive model is crucial; while the Coulomb friction law is common, more sophisticated models are necessary to represent real-world complexities, including surface roughness and temperature dependence. This review paper delves into the future research that prioritizes developing computationally efficient algorithms and parallel computing strategies. Advancements in constitutive modelling are vital for improved accuracy, along with enhanced contact detection algorithms for complex geometries and large deformations. Integrating experimental data and multiphysics capabilities will further enhance the reliability and applicability of these numerical techniques across various engineering applications. These advancements will ultimately improve the predictive power of simulations in diverse fields.

1. Introduction

Frictional contact problems are pervasive in engineering and scientific disciplines, encompassing a wide array of applications from the design of mechanical components and robotic systems to the modelling of geological processes and biological tissues [1]. These problems involve the interaction of deformable bodies, where forces are transmitted across interfaces subject to frictional constraints [2,3,4]. Accurately predicting the behavior of such systems is crucial for ensuring safety, performance, and efficiency [5,6]. The Stribeck curve, as shown in Figure 1, represents the graphical representation of the relationship between the coefficient of friction (μ) and the dimensionless parameter known as the Stribeck number (Sb) [7]. It describes the frictional behavior of lubricated contacts. The curve shows how friction transitions from boundary lubrication to mixed lubrication and, finally, to hydrodynamic lubrication as the Stribeck number increases [8]. However, the inherent nonlinearity and nonsmoothness associated with frictional contact phenomena present significant analytical and computational challenges. Closed-form solutions are typically unavailable, except for highly simplified scenarios, necessitating the development and application of robust numerical techniques [9].

This review focuses on the diverse numerical methods employed to address the complexities of frictional contact analysis. The fundamental challenge lies in the coupled nature of the problem: the contact conditions (whether contact exists and the magnitude and direction of contact forces) are themselves unknowns that depend on the deformation of the bodies involved [11]. This creates a system of equations that is nonlinear, often discontinuous, and potentially ill-conditioned. The presence of friction further complicates matters, introducing additional nonlinearities through Coulomb’s law, which relates frictional forces to the normal contact force and a coefficient of friction [12,13].

Several numerical approaches have been developed to tackle these challenges, each with its strengths and limitations. These broadly fall under the categories of finite element methods (FEM) [14,15], boundary element methods (BEM) [16,17], and discrete element methods (DEM) [18,19]. Within FEM, various strategies exist for handling contact constraints, including penalty methods, Lagrange multiplier methods, and augmented Lagrangian methods [20,21]. Each approach involves a different way of enforcing the contact and friction conditions within the governing equations [22]. Penalty methods introduce artificial compliance at the contact interface, while Lagrange multiplier methods introduce additional variables to represent the contact forces. Augmented Lagrangian methods combine aspects of both penalty and Lagrange multiplier methods, aiming to improve robustness and accuracy [23,24,25].

Furthermore, the choice of numerical integration scheme and solution algorithm significantly influences the accuracy and efficiency of the simulation. Implicit methods, often preferred for their stability, can require computationally intensive iterative solvers to handle the nonlinear system of equations [26,27]. Explicit methods, while computationally cheaper per timestep, may require smaller timesteps to ensure stability, especially in dynamic contact problems [21,28]. The selection of appropriate numerical techniques must consider factors, such as problem size, computational resources, desired accuracy, and the specific characteristics of the contact and friction conditions [29]. This review will explore these various aspects in detail, examining the theoretical foundations, practical implementations, and comparative advantages and disadvantages of each method. We will also address ongoing research directions aimed at improving the efficiency, accuracy, and robustness of frictional contact analysis.

1.1. The Importance of Frictional Contact Analysis in Engineering

Frictional contact analysis plays a crucial role in numerous engineering disciplines, impacting the design, performance, and safety of a vast array of systems [30]. Its importance stems from the pervasive nature of contact and friction in real-world engineering applications [31]. Neglecting these phenomena can lead to inaccurate predictions, compromised designs, and potentially catastrophic failures [32].

Frictional contact analysis is not just a specialized niche within engineering; it is a fundamental tool for accurately modelling and designing a wide range of systems [33]. Its application is crucial for ensuring safety, performance, and efficiency across diverse engineering disciplines. As computational power continues to increase, and numerical methods become more sophisticated, the importance of frictional contact analysis will only grow further [34].

1.2. Challenges in Modelling Frictional Contact Problems

Modelling frictional contact problems presents a formidable challenge due to the inherent complexities of the physical phenomena involved. These challenges stem from the nonlinear, nonsmooth, and often discontinuous nature of contact and friction, demanding sophisticated numerical techniques and careful consideration of various factors.

• The most significant challenge arises from the inherent nonlinearity of contact and friction. The contact conditions themselves are unknown a priori and depend on the deformation of the contacting bodies. This leads to a highly nonlinear system of equations [35,36].
• Accurately detecting contact between bodies is crucial. This involves determining which parts of the surfaces are in contact and calculating the penetration depth. This step is particularly challenging in large-scale simulations with complex geometries [37,38].
• The complexity is further compounded by the frictional behavior at the contact interface. The coefficient of friction is often not a constant but may vary with factors, such as surface roughness, temperature, velocity, and pressure. Modelling these variations accurately can be challenging, requiring sophisticated constitutive models and potentially experimental data to calibrate parameters [26,39].
• Solving frictional contact problems often involves significant computational costs. The nonlinearity of the problem necessitates iterative solution procedures, which can be computationally expensive, especially for large-scale models with many degrees of freedom [40].
• The choice of material model significantly influences the accuracy and computational efficiency of the simulation. Realistic materials often exhibit complex constitutive behavior, including plasticity, viscoelasticity, and damage. Incorporating these effects into the contact model adds significant complexity and computational cost [41,42].
• Microslip’s localized nature demands extremely fine meshes, increasing computational cost. Macroslip introduces path dependence and nonlinearity, requiring robust solvers. Modeling fretting and sliding wear necessitates coupling contact mechanics with material removal, involving empirical parameters and uncertainties in surface roughness and material properties. Accurately predicting the transition between micro- and macroslip remains challenging [43,44].
• Coupled thermo-mechanical effects in contact problems pose significant challenges to numerical analysis. Heat generation due to friction alters material properties, influencing contact pressure and wear. Accurately modeling this feedback loop requires solving coupled heat transfer and mechanical equations, increasing computational complexity. Precise material models accounting for temperature-dependent properties and wear mechanisms are crucial but often lack experimental validation, introducing uncertainties into the analysis [45,46].
• Transient wear problems, involving time-dependent changes in surface geometry and contact conditions, require computationally expensive time-stepping schemes. Steady-state analyses, while simpler, may not be applicable for many wear scenarios, particularly those involving significant material removal or changing contact pressures. Accurately capturing the transition between transient and steady-state behaviors presents a major challenge. Model validation against experimental data is crucial [47,48,49].

Overcoming these challenges requires a combination of advanced numerical techniques, robust algorithms, and careful consideration of the specific characteristics of the contact and friction phenomena. Ongoing research focuses on developing more efficient and accurate methods for modelling frictional contact, tackling issues like adaptive mesh refinement, multiscale modelling, and model order reduction.

1.3. Overview of Numerical Methods for Contact Analysis

Numerical methods are essential for solving frictional contact problems, because analytical solutions are generally unavailable except for highly simplified cases. Several approaches have been developed, each with its strengths and weaknesses. These can be broadly categorized into the finite element method (FEM), h-p version of FEM [50], penalty method, Lagrange multiplier method, augmented Lagrangian method, boundary element method (BEM), discrete element method (DEM), smooth particle hydrodynamics (SPH), and finite volume method (FVM) [51,52].

The choice of numerical method depends on several factors, including the problem’s geometry, material properties, and the desired level of accuracy and computational efficiency. Often, a combination of methods or a hybrid approach might be necessary to accurately and efficiently solve complex frictional contact problems. For example, a coupled FEM-DEM approach could be employed to model the interaction between a deformable body and granular material. Regardless of the chosen method, the careful consideration of contact detection algorithms and appropriate constraint enforcement techniques is crucial for obtaining reliable results.

1.4. Scope and Organization of the Review

This review aims to provide a comprehensive overview of numerical techniques employed in frictional contact analysis, encompassing both the theoretical foundations and practical applications of various methods. The scope extends across different numerical approaches, addressing their strengths, weaknesses, and suitability for various types of contact problems. The organization is structured to facilitate a clear understanding of the subject matter, progressing from fundamental concepts to advanced techniques. The remaining of the paper is organized in the following subsections:

The review’s organization aims for a logical progression of ideas, building upon fundamental principles to explore advanced techniques and applications. This structured approach ensures a comprehensive and accessible understanding of the state-of-the-art numerical methods for frictional contact analysis.

2. Fundamentals of Frictional Contact Mechanics

Frictional contact mechanics deals with the interaction between deformable bodies in contact, where forces are transmitted across the interface subject to frictional constraints. Understanding the fundamentals is crucial for developing and applying accurate numerical methods for contact analysis. The fundamentals of the frictional contact mechanics are described by the following laws:

2.1. Constitutive Laws for Friction

Constitutive laws for friction describe the relationship between the frictional force and the relative motion (or velocity) at the contact interface [53]. Several models exist, each with varying degrees of complexity and applicability:

• Coulomb friction: Coulomb friction is a widely used model describing the frictional force between two surfaces in contact. It states that the frictional force is proportional to the normal force, with the proportionality constant being the coefficient of friction (μ) [54,55]. The frictional force opposes relative motion and its magnitude is less than or equal to μ times the normal force [56,57]. When the tangential force reaches this limit, a slip occurs. While simple, Coulomb friction neglects velocity dependence and the transition between static and kinetic friction, limiting its accuracy in certain scenarios.
• Rate-dependent friction models: Rate-dependent friction models acknowledge that frictional forces are not solely determined by the normal force but are also significantly influenced by the relative sliding velocity between contacting surfaces [58,59,60]. Unlike the simplistic velocity-independent Coulomb friction model, rate-dependent models incorporate velocity as a key variable. Common examples include viscous friction models, which add a velocity-proportional term, and the Stribeck curve, which empirically describes the friction coefficient’s decrease at low velocities before levelling off at higher velocities. These models enhance the realism of simulations, particularly when dealing with phenomena like stick-slip motion or high-speed contacts, offering more accurate predictions of dynamic behavior than Coulomb friction alone. However, they introduce additional parameters requiring careful calibration [61].

The choice of friction model depends on the specific application and the desired level of accuracy. For many engineering applications, Coulomb friction provides a reasonable approximation. However, for situations where velocity dependence, surface roughness, or other factors are significant, more sophisticated rate-dependent or micro-mechanical models may be necessary. The model’s complexity must be balanced against computational cost and the availability of relevant material parameters.

2.2. Contact Kinematics and Geometry

Contact kinematics and geometry are fundamental aspects of frictional contact mechanics, dealing with the description of the relative motion and geometric relationships between contacting bodies. The accurate representation of these aspects is crucial for effective contact analysis [62].

• Contact detection: This is the initial and often computationally demanding step in contact analysis. It involves identifying which parts of the body are in contact. The methods used depend heavily on the dimensionality (2D or 3D) and the complexity of the geometries involved. Common approaches include node-to-surface, surface-to-surface, bounding volume methods, and mesh-based methods [63].
• Penetration depth: Once contact is detected, the penetration depth needs to be determined. This represents the amount by which one body penetrates another. The accurate calculation of penetration depth is crucial for calculating contact forces. Different methods exist to calculate penetration depth, depending on the contact detection method used [64].
• Contact area: The contact area is the region where the bodies are in physical contact. It can range from a point contact to a large, extended area depending on the geometry, loading conditions, and material properties. For simple geometries, the contact area can be easily determined. However, for complex geometries, numerical methods are often necessary to calculate the contact area [65].
• Relative motion: Describing the relative motion between the contacting bodies is critical for determining whether contact is sticking or slipping. This involves calculating relative velocity and relative displacement [66].
• Geometric considerations: Several geometric factors such as curvature, surface roughness, and geometric nonlinearities significantly influence contact analysis [67].

The accurate representation of contact kinematics and geometry is vital for accurate and reliable contact simulations. Efficient algorithms and appropriate numerical techniques are required to handle these aspects effectively, especially in large-scale simulations with complex geometries. Sophisticated contact detection and penetration depth calculations are essential for obtaining accurate results in frictional contact analysis.

3. Mathematical Formulation of Frictional Contact Problems

The mathematical formulation of frictional contact problems involves expressing the governing equations and constraints in a form suitable for numerical solution. This often involves the use of weak forms and variational inequalities.

• Governing equations: The foundation lies in the equations of motion (for dynamic problems) or equilibrium (for static problems) for the bodies involved. For small deformations, these are typically expressed using linear elasticity. For larger deformations, nonlinear elasticity or plasticity models may be necessary. These equations can be represented in various forms, including strong form and weak form.
• Contact and friction conditions: The contact and friction conditions at the interface are expressed mathematically as constraints:
  • Nonpenetration constraint: This constraint ensures that the bodies do not interpenetrate. It can be expressed using inequality conditions relating to the distance between the bodies at the contact interface.
  • Friction constraint: Coulomb’s law of friction, or other friction models, defines the relationship between the tangential force and relative velocity at the contact interface. This introduces a nonsmooth and nonlinear relationship, further complicating the problem.
• Variational inequalities: The combination of the weak form of the governing equations and the contact and friction constraints leads to a variational inequality formulation. This is a powerful mathematical framework for expressing and solving frictional contact problems. The variational inequality formulation encapsulates the entire problem, including the governing equations and all constraints. It provides a rigorous mathematical framework for analyzing and solving frictional contact problems.
• Discretization: To obtain a numerical solution, the variational inequality is discretized. This typically involves using finite element methods (FEM), dividing the domain into a mesh of elements and approximating the displacement field using shape functions. The variational inequality then becomes a system of nonlinear equations or inequalities, which can be solved using iterative methods, such as Newton–Raphson or conjugate gradient methods.
• Solution techniques: The solution of the resulting system of equations is often challenging due to the nonlinearity and nonsmoothness arising from the friction constraints. Various numerical methods are employed to handle this complexity, including penalty methods, Lagrange multiplier methods, and augmented Lagrangian methods.

The mathematical formulation using weak forms and variational inequalities provides a robust foundation for the numerical analysis of frictional contact problems. The choice of specific solution techniques heavily influences the efficiency and accuracy of simulations. The careful consideration of these aspects is crucial for reliable predictions in applications ranging from mechanical engineering to biomechanics.

4. Different Numerical Techniques Used in Frictional Analysis

4.1. Penalty Method

The penalty method is a common approach for solving constrained optimization problems, including those arising in frictional contact analysis. It approximates the constraints by adding penalty terms to the objective function. The penalty term is shown in the Equation (1).

$${\pi }_p={\displaystyle \frac{1}{2}}r{\left(g\right)}^2$$

(1)

where ${\pi }_p$ is the penalty function, $r$ is the penalty parameter, and $g$ represents an objective function.

In the context of contact mechanics, these penalty terms penalize interpenetration between bodies. Instead of explicitly enforcing the nonpenetration constraint (that bodies cannot interpenetrate), the penalty method introduces a penalty term to the system’s energy [18]. This term increases the total energy of the system, as the bodies penetrate each other. The penalty term is proportional to the amount of penetration, making it energetically unfavorable for penetration to occur.

The penalty method is an approximation of the contact constraint and does not strictly enforce it. Some level of interpenetration is always present, even with a very large penalty parameter [68]. This might be acceptable in some cases, but for applications requiring precise contact enforcement, other methods like Lagrange multipliers or augmented Lagrangian methods are generally preferred [69].

Despite its limitations, the penalty method’s simplicity and ease of implementation make it attractive for certain applications, particularly when high accuracy is not paramount. The choice of penalty parameter is critical; it is often determined through numerical experimentation or by analyzing convergence behavior. The careful selection of the penalty parameter is crucial for obtaining reliable and accurate results within the context of the penalty method.

4.2. Lagrange Multiplier Methods

The Lagrange multiplier method is a powerful technique for solving constrained optimization problems, offering a more rigorous approach to enforcing constraints than penalty methods. In the context of frictional contact problems, it introduces additional variables (Lagrange multipliers) to represent the contact forces, explicitly enforcing the contact and friction conditions [18].

The method introduces Lagrange multipliers as additional unknowns to the problem. These multipliers represent the constraint forces (contact forces in this case). The constraints are incorporated into the problem formulation using the Lagrange multipliers, transforming the constrained optimization problem into an unconstrained one in a higher-dimensional space [70].

$$\mathcal{L}\left(x,\lambda \right)=f\left(x\right)+〈\lambda ,g\left(x\right)〉$$

(2)

The Lagrangian is defined as shown in Equation (2) for the function f,g; the notation $〈.,.〉$ denotes the inner product. The symbol $\lambda$ is known as the Lagrange multiplier.

The Lagrange multiplier method offers several advantages over the penalty method, such as it enforces the constraints exactly (in theory), unlike penalty methods that only approximate them. The Lagrange multipliers directly represent the contact forces, offering more accurate and reliable information. It also eliminates the need to choose an optimal penalty parameter, a significant challenge with penalty methods [71].

This method is also associated with some limitations, such as introducing Lagrange multipliers increases the size of the system of equations, potentially increasing computational cost. Solving the resulting system of equations can be computationally more challenging than with penalty methods, often requiring specialized solution techniques.

In summary, the Lagrange multiplier method provides a rigorous and accurate approach to solving frictional contact problems, offering significant advantages over penalty methods, especially when precise contact force information is essential. However, the increased complexity of implementation and solution must be considered.

4.3. Uzawa Algorithm

The Uzawa algorithm is an iterative method used to solve saddle-point problems, a type of optimization problem that arises frequently in constrained optimization and, in particular, in frictional contact problems solved using the Lagrange multiplier method. It is particularly well-suited for problems where the constraints are linear or can be linearized. The Uzawa algorithm offers many advantages, such as it is relatively easy to implement compared to other methods for solving saddle-point problems [72]. The algorithm decouples the solution of the primary variables from the update of the Lagrange multipliers. This can lead to significant computational efficiency, especially when solving for u involves complex simulations (such as those using finite element methods). This method is more efficient for large-scale problems when combined with preconditioners and appropriate relaxation parameters [73].

Despite all these methods, this method is also associated with certain limitations. For instance, the convergence rate is slow especially without a preconditioner or for poorly conditioned problems. The choice of the relaxation parameter significantly influences the convergence rate. The performance of this method is sensitive to the choice of the relaxation parameter.

The Uzawa algorithm and its variants provide efficient methods for solving saddle-point problems arising from frictional contact analysis using the Lagrange multiplier method. The proper selection of parameters (preconditioner and relaxation parameter) and consideration of their limitations are essential for successful application. The choice between variants depends on the specific problem characteristics and computational resources.

4.4. hp-Version of FEM

The hp-version of the FEM is a powerful technique that combines the advantages of both the h-version and the p-version. Let us break down what that means:

• h-version: This traditional approach refines the mesh by reducing the element size (h). Accuracy is improved by increasing the number of smaller elements.
• p-version: This approach keeps the mesh size constant but increases the polynomial degree (p) of the approximating functions within each element. Accuracy is improved by using higher-order polynomials to better represent the solution.

The hp-version cleverly combines these two strategies as shown in Figure 2. It adaptively refines the mesh and increases the polynomial degree in different regions of the domain. This allows for highly efficient and accurate solutions, especially for problems with complex geometries or solutions exhibiting localized features, like singularities or boundary layers. Under certain conditions, the hp-version can achieve exponential convergence rates, meaning the error decreases much faster than with the h-version alone. This is a significant advantage for achieving high accuracy with fewer degrees of freedom. The adaptive refinement strategy allows the method to focus computational effort where it is most needed. Regions with complex behavior become refined and higher-order polynomials, while simpler regions may require less refinement and lower-order polynomials. This optimizes computational cost. The combination of h and p refinement often leads to significantly more efficient solutions than either method alone, especially for problems with singularities or boundary layers. Complexity: Implementing and managing the adaptive refinement and polynomial degree selection can be significantly more complex than the h-version. While ultimately more efficient for many problems, the implementation and computational cost per element can be higher than the h-version, particularly for very high polynomial degrees.

The hp-version of the finite element method offers a powerful and efficient approach to solving a wide range of problems requiring high accuracy. Its adaptive nature allows it to focus computational resources where they are most needed, leading to significant improvements in efficiency compared to traditional h-version methods. However, the increased complexity requires sophisticated implementation and management strategies.

The advantages and disadvantages of different numerical techniques in frictional contact analysis are represented in

Table 1. Comparison of different methods in terms of their advantages and disadvantages.

Numerical Technique	Advantages	Disadvantages
Penalty Method	Relatively simple to implement; easily incorporated into existing FE codes.	Can lead to ill-conditioning for large penalty parameters; penetration may occur; parameter selection is crucial.
Lagrange Multiplier Method	Accurate constraint enforcement; no penetration; no penalty parameter needed.	More complex to implement; leads to larger systems of equations; and can suffer from instability.
Augmented Lagrangian Method	Combines advantages of penalty and Lagrange multiplier methods; improved stability.	More complex than the penalty method; requires careful parameter tuning.
Barrier Method	Handles inequality constraints effectively.	Can be computationally expensive; requires careful parameter selection to avoid ill-conditioning.
Nonlinear Complementarity Problem (NCP) Methods	Can be very accurate and efficient for certain problem classes.	Can be complex to implement; convergence can be sensitive to initial guess and problem parameters.
Node-to-Node Contact	Simple implementation; computationally efficient for small contact areas.	Can lead to inaccuracies in large deformation problems; may not accurately capture complex geometries.
Node-to-Segment Contact	More accurate than node-to-node for larger contact areas.	More computationally expensive than node-to-node.
Segment-to-Segment Contact	Most accurate representation of contact geometry.	Most computationally expensive; complex implementation.

.

5. Computational Aspects and Efficiency

Computational efficiency is a critical concern in frictional contact analysis, especially for large-scale problems common in engineering simulations. The computational cost is significantly influenced by the chosen numerical technique and its implementation. Methods like the finite element method (FEM) are widely used but can be computationally expensive due to the nonlinearity of contact and friction, often requiring iterative solution strategies [75]. The complexity scales with the number of contact nodes and iterations needed for convergence.

The efficiency of iterative solvers, such as Newton–Raphson, plays a crucial role. Preconditioning techniques can dramatically improve convergence rates and reduce computational time. The choice of contact algorithm also significantly impacts efficiency [76]. Penalty methods, while simple to implement, can be inefficient for problems with high friction coefficients, while Lagrange multiplier methods, although more accurate, often require more computational effort. Node-to-node contact algorithms are generally more efficient than surface-to-surface methods, particularly in scenarios involving many small contacts [77].

Adaptive mesh refinement can improve accuracy and efficiency by focusing computational resources on regions with high-stress concentrations or complex contact geometries. Furthermore, parallel computing techniques, employing multiple processors, can significantly reduce solution time for large-scale problems [78]. The selection of an appropriate numerical integration scheme is also vital, with higher-order methods often leading to better accuracy but potentially increased computational cost. Finally, algorithmic optimizations and efficient data structures within the chosen numerical method can further enhance computational performance. The optimal choice often involves a trade-off between accuracy, computational cost, and the specific characteristics of the problem.

6. Applications and Case Studies

Frictional contact analysis is crucial across various engineering disciplines, enabling accurate modelling and the simulation of systems involving interacting bodies. Its applications span diverse engineering disciplines, offering valuable insights into system behavior and performance. Frictional contact analysis in rolling bearing as represented in Figure 3 is one of the applications.

In structural mechanics, frictional contact analysis is vital for predicting the load-bearing capacity and stability of structures. Examples include analyzing bolted joints, predicting the behavior of masonry structures, and modelling the interaction between soil and foundations. The accurate modelling of friction is crucial for assessing the safety and durability of these structures under various loading conditions. For instance, Cornejo et al. [80] introduced a framework that effectively models frictional contact in composite materials, particularly rubber–fiber composites, addressing incompressibility and fibred stiffening. The dual augmented Lagrangian and mortar methods ensure accurate contact force estimation, while the modified serial–parallel rule of mixtures handles diverse material behavior. Kwon et al. [4] used the NTS-AR method to accurately analyze 3D dynamic frictional contact under large deformation and plasticity, overcoming the limitations of traditional penalty-based NTS methods. Validation examples demonstrate its improved accuracy and applicability to complex scenarios like aircraft carrier landings. Wang et al. [44] provide a novel temperature-dependent model and experimental setup for accurate mechanical friction parameter identification in thermal environments, validated by minimal discrepancies between theoretical and experimental results.

Within machine design, understanding frictional contact is critical for optimizing the performance of mechanical components [33]. This includes analyzing the efficiency of gear trains, predicting wear and tear in bearings and bushings, and designing effective braking systems [81]. Simulations can help engineers optimize component designs for reduced friction, improved efficiency, and extended lifespan [82].

Biomechanics also benefits greatly from frictional contact analysis. Modelling the contact between bones and joints, such as in knee or hip replacements, is crucial for assessing implant design, predicting wear rates, and improving the overall performance of prosthetic devices [83]. Similarly, understanding the frictional forces in musculoskeletal systems is important for analyzing movement and injury prevention [84].

In manufacturing processes, frictional contact analysis contributes to the optimization of techniques like metal forming, machining, and sheet metal stamping. Accurate simulations can help predict material flow, stresses, and wear, allowing for improved process design and optimized tool geometries [85,86].

Finally, tribology, the study of interacting surfaces in relative motion, relies heavily on frictional contact analysis to study wear, lubrication, and friction reduction strategies. Simulations are instrumental in developing new materials and lubrication techniques for improved component efficiency and durability [87,88]. In summary, frictional contact analysis is an indispensable tool for understanding and predicting the behavior of a wide variety of mechanical systems.

These examples showcase the broad applicability of frictional contact analysis. The specific challenges and the required level of detail vary greatly, depending on the specific application. Selecting appropriate numerical techniques and material models is crucial for obtaining meaningful and reliable results.

7. Challenges of the Numerical Techniques in Frictional Contact Analysis

A review of numerical techniques for frictional contact analysis reveals several persistent challenges and exciting avenues for future research. One major hurdle is the inherent nonlinearity of frictional contact problems. The contact conditions themselves are nonlinear, changing dynamically as the system deforms, leading to complex iterative solution procedures that can be computationally expensive and prone to convergence difficulties. This is particularly pronounced in large-scale simulations with many contact interfaces. Furthermore, the Coulomb friction law, while widely used, is an idealized model and does not fully capture the complexities of real-world frictional behavior. More sophisticated constitutive models are needed, but incorporating them introduces further computational complexity.

Another key challenge lies in accurately modeling contact between dissimilar materials, especially at the microscale, where surface roughness and asperities play a significant role. Current numerical methods often struggle to capture these effects accurately, leading to potential inaccuracies in predicted contact forces and stresses. The development of multiscale methods that bridge the gap between macro- and microscale behavior is a promising area of future research.

The computational cost associated with frictional contact analysis remains a significant barrier, particularly for complex three-dimensional problems. The development of more efficient algorithms and parallel computing strategies is essential. This includes the exploration of advanced iterative solvers, preconditioning techniques, and adaptive mesh refinement to optimize computational performance. The integration of machine learning techniques to predict contact behavior or accelerate the convergence of iterative solvers offers a potential avenue for efficiency improvements.

8. Conclusions and Future Direction

The analysis of frictional contact problems remains a significant challenge in computational mechanics, demanding robust and efficient numerical techniques to accurately capture the complexities of contact interactions. This review has explored a range of established and emerging methods, highlighting their strengths and limitations in addressing the inherent nonlinearities and computational demands inherent in these problems. While significant progress has been made, several key areas require further investigation and development to enhance the accuracy, efficiency, and applicability of these techniques.

The choice of numerical method is often dictated by the specific characteristics of the problem, including the geometry, material properties, and the level of detail required. Finite element methods (FEM), due to their versatility and ability to handle complex geometries, remain the dominant approach. However, the nonlinearity introduced by contact and friction necessitates iterative solution procedures, often leading to significant computational costs, especially for large-scale problems. This highlights the need for the continued development of efficient solvers, including preconditioning techniques and parallel computing strategies, to mitigate these computational burdens. The exploration of alternative numerical methods, such as the boundary element method (BEM) or meshless methods, may offer advantages in specific scenarios, but their applicability to general frictional contact problems requires further investigation.

The accuracy of numerical simulations is critically dependent on the constitutive models employed to represent the frictional behavior of contacting surfaces. While the Coulomb friction law provides a widely used and relatively simple framework, it is an idealized model that may not accurately capture the complexities of real-world frictional interactions. More sophisticated constitutive models, incorporating factors such as surface roughness, adhesion, and temperature effects, are essential for achieving higher fidelity simulations. However, the incorporation of these advanced models often increases the computational complexity, demanding the further refinement of numerical techniques and algorithms.

The accurate representation of contact conditions themselves remains a critical area of focus. Penalty methods, while computationally efficient, can suffer from inaccuracies, particularly at high friction coefficients. Lagrange multiplier methods offer greater accuracy but often require more computational effort. The development of hybrid methods, combining the advantages of different approaches, presents a promising avenue for improving both accuracy and efficiency. Furthermore, the development of adaptive mesh refinement techniques, focusing computational resources on areas of high-stress concentration or complex contact geometries, can significantly enhance the accuracy and efficiency of simulations.

9. Future Trends and Research Opportunities

The field of numerical techniques for frictional contact analysis is ripe with opportunities for future research and development, driven by the increasing demand for accurate and efficient simulations in diverse engineering applications. Several key trends and research avenues are particularly promising:

• Advanced constitutive models: Current models often oversimplify frictional behavior. Future research should focus on developing more sophisticated constitutive models that incorporate factors, such as surface roughness, adhesion, temperature-dependent friction, and material anisotropy. This includes incorporating advanced experimental techniques to characterize frictional behavior at the micro- and nanoscale, informing the development of more realistic material models. Multiscale modelling approaches that bridge the gap between micro- and macroscale behavior will be crucial for accurately simulating complex contact scenarios.
• Enhanced computational efficiency: The computational cost of solving large-scale frictional contact problems remains a significant bottleneck. Future research should explore advanced numerical algorithms and parallel computing strategies to improve efficiency. This includes developing more efficient iterative solvers, preconditioning techniques, and adaptive mesh refinement strategies that dynamically adjust computational resources based on the problem’s complexity. The exploration of novel numerical methods, such as meshless methods or fast multipole methods, could also offer computational advantages for specific applications. The integration of machine learning techniques to accelerate convergence or even predict contact behavior directly represents a promising avenue for significant performance gains.
• Improved contact detection algorithms: Accurate and efficient contact detection is crucial for the success of frictional contact simulations. Future research should focus on developing more robust and efficient algorithms that can handle complex geometries and large deformations. This includes investigating advanced spatial data structures and algorithms that can quickly identify contacting elements, even in dynamic simulations with significant relative motion between bodies.
• Multibody dynamics and large deformations: Many real-world applications involve multiple interacting bodies undergoing large deformations. Future research needs to focus on developing robust numerical methods capable of handling these complex scenarios efficiently. This requires advancements in both contact detection and the solution of the governing equations of motion.
• Integration of experimental data and validation: The accuracy of numerical simulations depends heavily on the fidelity of input parameters and models. Future research should emphasize the integration of experimental data obtained from advanced experimental techniques, like digital image correlation (DIC) and micro-computed tomography (micro-CT), to validate and calibrate numerical models. This will lead to more accurate and reliable simulations.
• Coupling with other physical phenomena: Many real-world contact problems involve coupling with other physical phenomena, such as heat transfer, fluid flow, or wear. Future research should focus on developing multiphysics simulation capabilities that can accurately capture these coupled interactions.
• Application-specific methodologies: Developing specialized numerical techniques tailored to specific application domains (e.g., tire-road interaction, metal forming, biomechanics) is another important direction. This involves focusing on the specific challenges and complexities associated with each application, leading to more accurate and efficient simulations for those particular scenarios.

By pursuing these research directions, the field of numerical techniques for frictional contact analysis can continue to advance, leading to more accurate, efficient, and reliable simulations with a broader range of applications across various engineering disciplines.