A Concurrent Multiscale Framework for Concrete Damage Analysis Using Overlapping Domain Decomposition

Abstract

Failure of concrete structures is a multiscale process where macroscale at the structural level and mesoscale at the heterogeneous material level are both involved. A multiscale approach is necessitated in the simulation of concrete failure. Based on an overlapping domain decomposition method, a concurrent multiscale framework for the damage analysis of concrete structures is formulated. The applicability of the proposed framework is illustrated by the multiscale damage analysis of an L-shaped concrete structure. Considering the complexity of a mesoscale model for a global concrete structure, the concrete structure is divided into three parts that require different strategies. Special attention is paid to the part where mesoscale structure needs to be taken. The Concrete Damaged Plasticity (CDP) model is adopted at the mesoscale level. The numerical results indicate that the proposed framework is able to model the damage process in concrete structure where a critical area will be particularly considered. The computational efficiency of the concurrent nonlinear algorithm is also discussed. The proposed multiscale framework can be potentially applied to model structural damage analysis in engineering practice.

1. Introduction

Concrete structures are one of the most common types of structures around the world. Throughout their service life, concrete structures may be subjected to a variety of loads that cause structural degradation and damage. In recent decades, structural problems related to failure of concrete structures have increased drastically. Generally, concrete is treated as homogeneous material that is assumed to be linear elastic. However, concrete behavior at the mesoscale is more complex, and the heterogeneous internal material structure cannot be ignored [1]. The size of material components such as aggregate is much smaller compared with the structural level. There are at least two scales involved, namely, macroscale and mesoscale. Damage localization at the mesoscale may cause the formation of macroscopic cracks. Thus, failure of concrete structures is a multiscale process [2,3]. Though accurate numerical results benefit from adopting detailed fine-scale models that account for material heterogeneities, modeling large-scale structures is computationally not feasible. Therefore, studies have been conducted to develop multiscale computational frameworks [4,5,6] with reasonable computational cost, as well as satisfactory accuracy of the mechanical model.

A multiscale framework can be mainly summarized as hierarchical multiscale models and concurrent multiscale models. The former basically consider and solve macroscale and mesoscale problems independently, and computation information is passed through different scales. The latter consider that the problem domain is decomposed into several subdomains that are treated by different strategies, such as FEM and a more fine-scale model. These subdomains are connected via appropriate constraints, and computations are performed concurrently. Over the past decades, there have been considerable efforts aimed at developing both hierarchical and concurrent multiscale models [7].

A hierarchical multiscale approach is proposed to evaluate the nonlinear constitutive behavior of concrete reinforced with carbon nanotubes in [8]; it is noted that a hierarchical multiscale analysis strategy is implemented to pass information through scales with different RVEs assigned at each scale of separation. A multiple-time-scale algorithm is developed in [9]; different length scales are adopted and correspond to different time steps. The computations are performed independently, and thus are also categorized as a hierarchical multiscale approach. A multiscale constitutive model for fiber-reinforced concrete is developed in [10], which homogenizes the microscale behavior of fibers, cemented aggregate particles, and air voids across a representative volume element.

Concurrent multiscale modeling of three-dimensional crack propagation in concrete is established in [11]; a macroscopic model with homogenized elastic parameters is employed in the regions where the material behaves elastically, while a mesoscopic model based on a mesh fragmentation technique is adopted to describe the concrete as a heterogeneous three-phase material composed of a mortar matrix, coarse aggregates, and interfacial transition. A dynamic FEM-DEM multiscale modeling approach for concrete structures is developed in [12], which couples the macroscale finite element method and the mesoscale discrete element method to capture the strain rate effects of concrete. It will be noted that an appropriate tensile damage constitutive model is adopted to describe the process of crack initiation and propagation. For fiber-reinforced concrete, dynamic constitutive models within damage frameworks have been established to characterize the damage evolution process [13]. To name just a few, a variety of multiscale modeling approaches can be found in [14,15,16,17,18,19]. Recent advances on experimental technology, such as optical technique, is used for multiscale damage assessment of concrete [20], and a digital image correlation method is adopted for mapping multiscale damage in concrete [21].

To address the computational challenges in concurrent multiscale modeling, machine learning methods have been introduced to accelerate multiscale analysis [22], which significantly reduces the computational cost by replacing complex microscopic mechanical solution problems with trained neural networks. Deep learning surrogates have also been developed for phase field modeling in fiber-reinforced composites, achieving exceptional prediction accuracy while providing several orders of magnitude acceleration compared with conventional phase field computations [23]. Physics-informed machine learning approaches have been systematically applied to structural failure analysis, providing a framework that integrates physical principles with data-driven methods to enhance accuracy and interpretability in damage prediction and failure mechanism analysis [24].

Within the current literature on the multiscale analysis of concrete materials and structures, it is noteworthy that a hybrid multiscale framework is formulated based on domain decomposition techniques in [25,26]. The proposed framework can be used for the analysis of non-linear heterogeneous materials such as concrete, and is capable of tackling strain localization and failure phenomena. Domain decomposition techniques are used to partition the material in a number of non-overlapping domains. Particularly, adaptive refinement is performed at those domains affected by damage processes. It will be pointed out that domain decomposition methods are efficient and flexible. More importantly, such methods are inherently suitable for parallel computing [27]. Compared with non-overlapping domain decomposition approaches that require explicit treatment of interface conditions between adjacent subdomains, the overlapping formulation avoids the complexity of interface coupling by allowing subdomains to share a common region where solutions are reconciled through an iterative process. This feature is particularly advantageous for multiscale problems where the transition between different modeling strategies (e.g., from mesoscale damage models to macroscale elastic models) needs to be handled smoothly without introducing artificial stress discontinuities at the interfaces. For large-scale structural problems such as progressive collapse analysis [28,29], where failure initiates at local critical regions and propagates through the structure, modeling the entire structure at the mesoscale level is computationally prohibitive. The overlapping domain decomposition approach provides a natural and efficient solution by confining the computationally expensive mesoscale modeling to the critical regions while treating the remainder of the structure at the macroscale with homogenized properties, thereby achieving a significant reduction in computational cost without sacrificing accuracy in the damage zone. Though literature on multiscale methods for the analysis of concrete structures has been enormous, a multiscale framework based on a domain decomposition method is rather limited except for the aforementioned work. Efforts on domain decomposition methods are mainly taken for the solution of mathematical problems such as partial differential equations [30,31].

In our previous work [32,33,34,35], numerical modeling on concrete materials and structures is studied, emphasizing crack initiation and propagation, as well as crack formation and coalescence. The present work primarily aims to develop a concurrent multiscale framework for the damage analysis of concrete structures on the basis of an overlapping domain decomposition method. The Concrete Damaged Plasticity (CDP) model is employed to describe the process of damage evolution. The idea is to propose a practical and efficient model that can be implemented in commercial software such as Abaqus. In this regard, special emphasis is placed on algorithm implementation and computational efficiency. The overlapping domain decomposition approach offers a natural mechanism for bridging the mesoscale and macroscale representations: the overlapping region acts as a transition buffer that ensures smooth information transfer between different modeling strategies without requiring complex coupling conditions. This makes the framework both conceptually straightforward and practically implementable within existing commercial finite element platforms.

This paper is structured as follows: The multiscale framework for damage analysis of concrete structures is formulated in Section 2. Section 3 presents the numerical case of an L-shaped concrete structure, as well as the damage model adopted in the modeling. Results and discussions are found in Section 4. The main conclusions of this paper are summarized in the end, and perspectives are discussed.

2. Methods

2.1. Damage Constitutive Model for Concrete

The Concrete Damaged Plasticity (CDP) model has been proven a successful constitutive approach for concrete. In theory, the total strain tensor $\mathit{\epsilon }$ could be decompressed by the elastic part ${\mathit{\epsilon }}^{\mathrm{el}}$ and the plastic part ${\mathit{\epsilon }}^{\mathrm{pl}}$.

$$\mathit{\epsilon }={\mathit{\epsilon }}^{\mathrm{el}}+{\mathit{\epsilon }}^{\mathrm{pl}}$$

(1)

By introducing the damage variable d (scalar) and the effective stress tensor $\overline{\mathit{\sigma }}$ [36,37],

$$\mathit{\sigma }=(1-d)\overline{\mathit{\sigma }},\overline{\mathit{\sigma }}\triangleq {\mathit{D}}_0^{\mathrm{el}}:(\mathit{\epsilon }-{\mathit{\epsilon }}^{\mathrm{pl}})$$

(2)

the stress–strain relationship could be derived, as follows:

$$\mathit{\sigma }={\mathit{D}}^{\mathrm{el}}:(\mathit{\epsilon }-{\mathit{\epsilon }}^{\mathrm{pl}})=(1-d){\mathit{D}}_0^{\mathrm{el}}:(\mathit{\epsilon }-{\mathit{\epsilon }}^{\mathrm{pl}})$$

(3)

Here, $\mathit{\sigma }$ is the Cauchy stress tensor. ${\mathit{D}}_0^{\mathrm{el}}$ and ${\mathit{D}}^{\mathrm{el}}$ are the initial and damaged stiffness tensor ${\mathit{D}}^{\mathrm{el}}$ of concrete, respectively, also ${\mathit{D}}^{\mathrm{el}}=(1-d){\mathit{D}}_0^{\mathrm{el}}$.

The CDP model assumes that concrete failure is mainly caused by tensile crack or compressional crushing, which is controlled by equivalent plastic strain ${\tilde{\epsilon }}_t^{\mathrm{pl}}$ (scalar) in tension and ${\tilde{\epsilon }}_c^{\mathrm{pl}}$ in compression. In uniaxial tension and compression, constitutive equations are respectively written as

$${\sigma }_t=\left(1-d_t\right)E_0\left({\epsilon }_t-{\tilde{\epsilon }}_t^{\mathrm{pl}}\right)$$

(4)

$${\sigma }_c=\left(1-d_c\right)E_0\left({\epsilon }_c-{\tilde{\epsilon }}_c^{\mathrm{pl}}\right)$$

(5)

where the subscripts t and c represent tension and compression, respectively, and the damage variables $d_t$ and $d_c$ are related to temperature, equivalent plastic strain, and other material constants.

For cyclic loading conditions, concrete shows some recovery ability of elastic stiffness as the load changes sign, especially as the load changes from tension to compression. This effect is mainly caused by crack closure. To describe this effect, the damage could be redefined as

$$(1-d)=(1-s_t{d}_c)(1-s_c{d}_t)$$

(6)

Degradation of elastic modulus could thus be written in a general form, as follows:

$$E=(1-d)E_0$$

(7)

where $E_0$ is the initial modulus. $s_t\in [0,1]$ and $s_c\in [0,1]$ are related to the state of stress reversal

$$s_t=1-w_t{r}^{*}\left({\sigma }_{11}\right),0\le w_t\le 1$$

(8)

$$s_c=1-w_c\left[1-r^{*}({\sigma }_{11})\right],0\le w_c\le 1$$

(9)

where

$$r^{*}\left({\sigma }_{11}\right)=H\left({\sigma }_{11}\right)=\left\{\begin{array}{ccc}1& \mathrm{if}& {\sigma }_{11}>0\\ 0& \mathrm{if}& {\sigma }_{11}<0\end{array}\right.$$

(10)

Additionally, $w_t$ and $w_c$ are weight factors controlling the recovery of stiffness upon sign change of the load.

In experimental and numerical practice, another two variables are usually used to describe the nonlinear stress–strain behavior—cracking strain ${\tilde{\epsilon }}_t^{\mathrm{ck}}$ in tension and inelastic strain ${\tilde{\epsilon }}_c^{\mathrm{in}}$ in compression. The relationship between these two variables and the equivalent plastic strain could be derived as

$${\tilde{\epsilon }}_t^{\mathrm{pl}}={\tilde{\epsilon }}_t^{\mathrm{ck}}-{\displaystyle \frac{d_t}{\left(1-d_t\right)}}{\displaystyle \frac{{\sigma }_t}{E_0}}$$

(11)

$${\tilde{\epsilon }}_c^{\mathrm{pl}}={\tilde{\epsilon }}_c^{\mathrm{in}}-{\displaystyle \frac{d_c}{\left(1-d_c\right)}}{\displaystyle \frac{{\sigma }_c}{E_0}}$$

(12)

Therefore, given the stress–strain relationship under uniaxial loading condition, material constants could be determined by the above expressions.

2.2. Overlapping Domain Decomposition Method for Multiscale Damage Analysis

In the present work, overlapping domain decomposition (ODD) methods are introduced to fulfill the multiscale damage analysis. The domain decomposition (DD) method is a numerical framework for solving PDE. Its advantages mainly lie on the capability of parallel computing, as well as using different numerical techniques, even different governing equations in different subdomains [27,38,39].

The ODD method falls into one special group of DD methods using overlapping subdomains, which has an iterative but simple algorithmic structure, avoiding treating interface problems between neighboring domains. Therefore, it can use multiple CPUs or PCs to solve each subdomain problem independently in each iteration. The size of the overlapping region is an important parameter that influences the convergence rate of the iterative process. In general, a larger overlap facilitates faster convergence by providing more shared information between neighboring subdomains, but at the cost of increased computational effort due to the redundant computation in the overlapping zone. The appropriate balance between overlap size and computational efficiency is problem dependent and is investigated in the numerical study presented in Section 3.

In this paper, the model for concrete damage analysis contains multiple subdomains at two scales. Mesoscopic substructures are usually located in areas of special concern. The selection of mesoscale subregions is guided by preliminary elastic analysis that identifies stress concentration zones and anticipated damage initiation areas. In the preliminary analysis, the entire structure is treated as a homogeneous elastic body, and the resulting stress field reveals the locations where high stress gradients and potential damage initiation are expected. These critical zones are then designated as mesoscale subdomains. This ensures that the fine-scale resolution is allocated to regions where damage is most likely to occur and where heterogeneous material behavior plays a dominant role, while the remaining regions are modeled at the macroscale to maintain computational efficiency.

From an energy perspective, the strain energy in the overlapping domain is given by $U_{{\mathsf{\Omega }}_{ov}}={\int }_{{\mathsf{\Omega }}_{ov}}{\displaystyle \frac{1}{2}}{\sigma }_{ij}{\epsilon }_{ij}dV$. Since the convergence of the additive Schwarz algorithm ensures the consistency of both displacement and stress fields within the overlapping region, the strain energy is also automatically consistent between the mesoscale and macroscale representations. This energy consistency is an inherent consequence of the algorithm’s convergence rather than an independently imposed constraint, which is a distinctive advantage of the overlapping formulation over non-overlapping approaches that typically require explicit energy-based interface coupling conditions.

From the classical ODD theory [38], the convergence behavior of the additive Schwarz method is characterized by a contraction factor $\rho <1$, such that $\parallel e^{n+1}\parallel \le \rho \parallel e^n\parallel$. For overlapping subdomains with overlap width $\delta$ and subdomain characteristic size H, the classical estimate gives $\rho \approx 1-C\delta /H$, where C is a positive constant. This indicates that a larger overlap leads to faster convergence, which is consistent with the numerical observations presented in Section 3.3. It should be noted that this classical estimate is rigorously derived for linear problems. For nonlinear problems involving damage evolution, while a rigorous theoretical proof remains an open question, the numerical experiments presented in Section 4 demonstrate that the framework converges reliably for the nonlinear damage problem considered in this work. The computational accuracy in these subdomains is ensured by introducing fine FE meshes. Additionally, the evolution of mesoscopic defects and the growth of cracks are described by the theory of continuum damage mechanics. Macroscopic substructures are arranged on the area far away from the hot spot, where the stress and strain are relatively non-significant, and/or damage evolution is slow or even retarded. Within a macroscopic domain, models are meshed into relatively coarse grids, and material is considered linear and elastic.

A classic ODD method originates from the Schwarz alternating method, and by nature are sequential rather than parallel. Here, we use a modified version of the Schwarz method called additive Schwarz method. It has an inherently parallel computing character, and thus suitable for concurrent multiscale analysis of damage evolution.

The global domain $\mathsf{\Omega }$ is decomposed into a set of $P+Q$ subdomains ${\left\{{\mathsf{\Omega }}_i\right\}}_{i=1}^{P+Q}$, which could be arranged in the order that the first P subdomains are macro-subdomains ${\left\{{\mathsf{\Omega }}_i\right\}}_{i=1}^P={\left\{{\mathsf{\Omega }}_i^{\mathrm{macro}}\right\}}_{i=1}^P$ and then the Q micro-subdomains ${\left\{{\mathsf{\Omega }}_i\right\}}_{i=P+1}^{P+Q}={\left\{{\mathsf{\Omega }}_i^{\mathrm{micro}}\right\}}_{i=1}^Q$, such that $\mathsf{\Omega }={\cup }_{i=1}^P{\mathsf{\Omega }}_i^{\mathrm{macro}}{\cup }_{j=1}^Q{\mathsf{\Omega }}_j^{\mathrm{micro}}$. Here, ${\mathsf{\Gamma }}_i$ denotes the internal (artificial) boundary of the subdomain i, i.e., the part not belonging to the global boundary $\partial \mathsf{\Omega }$.

In macroscale domains, the governing equation reads [40]

$$A^{\mathrm{macro}}\mathbf{u}=\mathbf{f}$$

(13)

where $A_{\mathrm{macro}}$ is a linear operator

$$A^{\mathrm{macro}}=-\mu \Delta -(\lambda +\mu )\mathrm{grad}\mathrm{div}$$

(14)

i.e.,

$$A^{\mathrm{macro}}\mathbf{u}=-\mu u_{i,kk}-(\lambda +\mu )u_{k,ki}$$

(15)

Here, $\lambda$ and $\nu$ are Lamé parameters.

Additionally, in micro domains,

$$A^{\mathrm{micro}}\mathbf{u}=\mathbf{f}$$

(16)

Here, the operator $A^{\mathrm{micro}}$ is nonlinear that

$$A^{\mathrm{micro}}\mathbf{u}={(D_{ijkl}^{\mathrm{el}}({\epsilon }_{kl}(\mathit{u})-{\epsilon }_{kl}^{\mathrm{pl}}))}_{,j}$$

(17)

$D_{ijkl}^{\mathrm{el}}$ is the damaged stiffness tensor as defined in Equation (3), and the displacement–strain relationship is

$${\epsilon }_{kl}(\mathit{u})={\displaystyle \frac{1}{2}}(u_{k,l}+u_{l,k})$$

(18)

Overall, the BVP on the global domain could generally be described as

$$\left\{\begin{array}{c}A\mathbf{u}=\mathbf{f}\mathrm{in}\mathsf{\Omega }\hfill \\ \mathbf{u}=\mathbf{g}\mathrm{on}\partial \mathsf{\Omega }\hfill \end{array}\right.$$

(19)

When it comes to a specific subdomain i, the restriction of A onto ${\mathsf{\Omega }}_i$, i.e., $A_i$, refers to $A^{\mathrm{macro}}$ or $A^{\mathrm{micro}}$. It should also be noted that, here, we just denote the problem of displacement boundary condition (Dirichlet condition); other kinds of boundary conditions are similar.

The algorithm for multiscale damage analysis is as follows in Algorithm 1. Here, the superscript n means the n-th iteration.

Algorithm 1 Multiscale damage analysis of concrete structure

Step 1. Initial guess of $u^0$, $n:=0$

Step 2. Solving BVP on subdomains ${\mathsf{\Omega }}_i$ concurrently ($i=1,2,\cdots ,P+Q$) $$\left\{\begin{array}{c}{A}_i{\mathbf{u}}_i^{n+1}=\mathbf{f}\mathrm{in}{\mathsf{\Omega }}_i\hfill \\ {\mathbf{u}}_i^{n+1}=\mathbf{g}\mathrm{on}\partial {\mathsf{\Omega }}_i\setminus {\mathsf{\Gamma }}_i\hfill \\ {\mathbf{u}}_i^{n+1}={\mathbf{u}}_i^n\mathrm{on}{\mathsf{\Gamma }}_i\hfill \end{array}\right.$$ (20)

Step 3. Interpolate ${\mathbf{u}}_i^{n+1}$ onto the global domain $\mathsf{\Omega }$, i.e., $${\tilde{\mathbf{u}}}_i^{n+1}=\left\{\begin{array}{c}{\mathbf{u}}_i^{n+1}x\in {\mathsf{\Omega }}_i\\ {\mathbf{u}}_i^{n}x\in \mathsf{\Omega }\setminus {\mathsf{\Omega }}_i\end{array}\right.$$ (21)

Step 4. Averaging $${\mathbf{u}}^{n+1}={\displaystyle \frac{1}{P+Q}}\sum _{i=1}^{P+Q}{\tilde{\mathbf{u}}}_i^{n+1}$$ (22)

Set $n:=n+1$ and go back to Step 2. Repeat until $${\displaystyle \frac{∥{\mathbf{u}}^{n+1}-{\mathbf{u}}^n∥}{∥{\mathbf{u}}^{n+1}∥}}<{ϵ}_t$$ (23)

Return ${\mathbf{u}}^{n+1}$

To improve the convergence rate, each subdomain could introduce a relaxation parameter [38,41] ${\theta }_i$ and $\theta ={\displaystyle \frac{1}{P+Q}}\sum {\theta }_i<2$, and Equation (22) is replaced by the following expression:

$${\mathbf{u}}^{n+1}=\sum _{i=1}^{P+Q}{\theta }_i{\tilde{\mathbf{u}}}_i^{n+1}+(1-\theta ){\mathbf{u}}^n$$

(24)

Computations on substructures are completely parallel, while the final synchronization/compatibility of displacement is realized through repetitively communicating and adjusting between neighboring subdomains. Due to the possible inconsistency of FE mesh in overlapped domains, FE-based piecewise interpolation is used for extending displacements from one subdomain to another.

2.3. Implementation

The commercial FEM software Abaqus 2020 is used as a platform to implement the algorithm as stated in the previous section. A schematic diagram is depicted in Figure 1.

Among all processes, communication between neighboring substructures is the key point in implementation consideration. It mainly includes mutually extracting nodal stresses and displacements from each other first, and after the process of interpolation (Step 3) and averaging (Step 4), applying the calculated displacements on the artificially internal boundaries as updated boundary conditions in the next iteration. On the Abaqus platform, there are two technical ways to fulfill the algorithm. The first one is to use the interface of user-defined subroutines. Since DLOAD and DISP subroutines could define specific force and displacement loading (changing with time and space), respectively, they could be used in combination. Using Python (Version 2.7.15 in Abaqus 2020) interface is anther choice. Via this interface, the modeling module and post-processing module in Abaqus could be called by Python language. Through the modeling module, the boundary conditions of subdomains could be updated. Additionally, through the post-process module, displacement and stress results could be extracted for further treating. Compared with user-defined subroutines, Python interface has much more advantages in coding productivity and portability that is much more important for engineering practice. Therefore, the second way is adopted in the present work.

In nonlinear damage analysis, particular attention must be paid to numerical stability when the damage variable d approaches 1, as the material stiffness $E=(1-d)E_0$ degrades to near zero, potentially leading to ill conditioning of the stiffness matrix and convergence difficulties. Several measures are adopted in this work to address this issue. First, the viscoplastic regularization provided by the Abaqus CDP model is employed by specifying a small viscosity parameter. This regularization allows the stress state to temporarily exceed the yield surface during the iterative solution, which effectively smooths out the rapid stiffness degradation and significantly improves the convergence behavior in regions with high damage gradients. Second, the upper limit of the damage variable is controlled to be slightly less than 1.0 (e.g., $d_{max}=0.99$) in the numerical implementation, thereby preventing the element stiffness from becoming completely zero and avoiding singularity of the global stiffness matrix. Third, the automatic time stepping algorithm in Abaqus plays an important role in maintaining convergence. When the nonlinear solver detects convergence difficulties within a load increment, the algorithm automatically reduces the increment size, allowing the material state to evolve more gradually and the equilibrium iterations to converge. This mechanism is particularly effective during the damage localization phase when the material response changes rapidly. Furthermore, it is worth emphasizing that, within the domain decomposition framework, the damage evolution is confined to the mesoscale subdomain, while the macroscale subdomains remain linear elastic. This spatial confinement of nonlinear behavior means that numerical instability is localized to the mesoscale region and does not propagate to or destabilize the macroscale solution, which is an inherent advantage of the proposed concurrent multiscale approach.

Another issue that should be noted here is the initial guess. In the nonlinear analysis evolving non-trivial damage evolution, zero initial guess may cause non-convergence, which differs from the situation of linear problems. Here, a pre-calculation process is suggested. The pre-calculation is an linear and elastic analysis on the structure, which means that this model is nearly the same as the multiscale damage model except the damage is not introduced in mesoscale subdomains. The linear problem should converge quickly (usually 10 iterations are enough). Additionally, the obtained displacement on internal boundaries could be used as the initial guess, which could effectively avoid the non-convergence caused by imposing unappropriated initial boundary conditions.

3. Multiscale Damage Analysis of Concrete: An Application

3.1. The Multiscale Model of L-Shaped Concrete Structure

The L-shaped concrete component is widely used in engineering structures such as foundation pit bracing and subgrade support and so on. In this section, a two-dimensional L-shaped component is chosen for damage analysis. Geometrical size and loading boundary conditions are shown in Figure 2, where the bottom is fully constrained, and the right side is subjected to a vertical displacement of 0.25 mm.

The multiscale model consists of two macroscale substructures and one mesoscale substructure. Through preliminary elastic analysis, the corner region of the L-shaped structure is identified as the critical zone exhibiting the highest stress concentration under the applied loading. Under the vertical displacement applied to the right limb, the inner corner acts as a geometric discontinuity that generates significant stress concentration, making it the most likely location for damage initiation and crack propagation. This region is therefore designated as the mesoscale subdomain for detailed damage analysis, where the heterogeneous composition of mortar and aggregates is explicitly modeled and the CDP constitutive model is employed to capture the nonlinear damage evolution. The two limbs away from the corner, where the stress levels remain well below the material strength, are treated as macroscale subdomains with homogenized elastic properties. As seen in Figure 2, material heterogeneity and mesoscale damage evolution will not be considered in macro domain. Additionally, material constants such as elastic modulus and Poisson’s ratio are calculated via a homogenization process. In mesoscale substructure, concrete is modeled as a composite of mortar and aggregates. For convenience, the aggregates are distributed uniformly in the area with a uniform size of 30 mm. The volume ratio of aggregates is 20%. Material parameters of the mortar and aggregate in the substructure are shown in

Table 1. Material constants.

Material	Constitutive Model	Parameter	Value
Mortar	CDP	Young’s modulus (GPa)	38
		Density (kg/m3)	2.75 × 103
		Poisson’s ratio	0.2
		Dilatancy angle (°)	30
		Stress ratio	1.16
		Tensile strength (MPa)	3.6
		Compressive strength (MPa)	4.5
		Compressive limit (MPa)	2.8
Aggregates	Elastic	Young’s modulus (GPa)	73
		Density (kg/m3)	2.75 × 103
		Poisson’s ratio	0.2
Concrete	Elastic	Young’s modulus (GPa)	45
		Density (kg/m3)	2.75 × 103
		Poisson’s ratio	0.2

.

The material parameters listed in Table 1 were adopted from established literature rather than being independently calibrated through experiments, as the primary focus of this work is the development and validation of the multiscale framework rather than precise material characterization. The mechanical properties of mortar, including Young’s modulus (38 GPa), tensile strength (3.6 MPa), and compressive strength (4.5 MPa), are typical values widely reported in the literature on mesoscale concrete modeling [36,37,42]. It should be noted that, in mesoscale modeling, “mortar” refers specifically to the cement paste matrix phase (excluding aggregates), which has lower strength than macroscopic concrete. The compressive strength of 4.5 MPa for the mortar matrix is consistent with the values reported in mesoscale modeling literature and represents the weaker matrix phase, whereas the aggregates are assigned a higher elastic modulus (73 GPa) to reflect their stiffer and stronger nature. The dilatancy angle of 30° is a well-accepted standard value for cementitious materials in the CDP framework. The biaxial-to-uniaxial compressive strength ratio of 1.16 originates from the classical biaxial tests by Kupfer et al. [43] and is the default value recommended in the Abaqus CDP model. The homogenized elastic modulus of concrete at the macroscale (45 GPa) was calculated through the rule of mixtures considering an aggregate volume fraction of 20%. The tensile and compressive damage evolution variables ($d_t$ and $d_c$) were not independently calibrated but were derived from the uniaxial stress–strain relationships through the standard CDP formulation described in Equations (4) and (5), following the damage plasticity theory proposed by Lee and Fenves [36]. These parameter values fall within well-established engineering ranges and are considered sufficient for demonstrating the effectiveness and robustness of the proposed multiscale framework.

3.2. Results

The fields of damage variable, stress, and displacement of the concrete structure are solved through multiscale modeling and the overlapped DD algorithm, as shown graphically in Figure 3.

Contour of displacements is plotted in Figure 3a,b, where the overlapped domains are boxed with dashed lines. It could be seen that structural displacement in the vertical direction is significantly larger than in the horizontal direction, and deformation in areas far from the corner roughly coincides with classical beam theory. It is noticed that both $u_x$ and $u_y$ are well consistent within overlapped areas, suggesting that the proposed DD-based concurrent damage analysis as well as its implementation strategy is correct. Additionally, finite/large deformation occurs inside the formed band of high damaged area (where E = 0), finally leading to displacement discontinuity along the “crack” surface. The smooth transition of displacement fields from the macroscale subdomains into the mesoscale subdomain through the overlapping region demonstrates that the additive Schwarz iterative scheme effectively reconciles the solutions from different scales. The displacement discontinuity observed along the damage band is a physical phenomenon associated with crack opening, rather than a numerical artifact of the domain decomposition, confirming the physical fidelity of the proposed multiscale approach.

Figure 3c–e show contours of the three stress components, i.e., ${\sigma }_x$, ${\sigma }_y$, and ${\tau }_{xy}$. It can be seen that the stress contours in overlapping areas are also consistent, which verifies the effectiveness of the method. Additionally, shear stress ${\tau }_{xy}$ is significantly smaller than normal stresses ${\sigma }_x$ and ${\sigma }_y$. The consistency of stress fields within the overlapping regions is a critical indicator of the accuracy of the proposed framework, as it demonstrates that the solution obtained from the concurrent multiscale analysis agrees with the solution that would be obtained from a single-scale analysis in these transitional zones. The dominance of normal stresses over shear stress in this loading configuration is consistent with the expected failure mode of tensile splitting at the inner corner of the L-shaped structure.

Figure 3f illustrates the contour of a damage variable. Since the constitutive relation based on continuum damage mechanics is not introduced in marco-domains, i.e., subdomains A and C, it only plots the distribution of a damage variable within subdomain C, where the areas filled with white color are aggregates. It is seen that, under vertical load, the main failure mode of concrete is tensile rupture. Damage in some area reaches its limit of 1.0, meaning that the elastic modulus in this localized area is reduced to zero. It also shows that the highly damaged area progresses to automatically form a “crack”. Due to the appearance of macro crack, damage in other areas is trivial. The crack initiates from the inner corner of the L-shaped structure where the stress concentration is most severe, and propagates through the mortar matrix along a path that is influenced by the spatial distribution of aggregates. The aggregates, being stiffer and stronger than the mortar, act as obstacles that deflect the crack path, resulting in a tortuous crack trajectory that is characteristic of mesoscale concrete fracture [44]. This realistic crack pattern demonstrates the capability of the concurrent multiscale framework to capture the essential features of mesoscale damage mechanisms that would not be visible in a purely macroscopic analysis.

It is worth noting that the damage is highly concentrated within the interior of the mesoscale subdomain and does not extend toward the meso–macro interface. This observation has important implications for the validity of the proposed framework. The overlapping region between the mesoscale and macroscale subdomains effectively serves as a transition buffer zone. Within this buffer zone, both the mesoscale and macroscale models coexist, and the displacement and stress fields are communicated and reconciled through the iterative process described in Algorithm 1. The high consistency of displacement and stress contours within the overlapping areas, as demonstrated in Figure 3a–e, confirms that the information transfer between the two scales is accurate and that the transition from the fine-scale nonlinear response to the coarse-scale elastic response is smooth. This confirms that the mesoscale region, as selected through the preliminary elastic analysis, is sufficiently large to fully contain the damage evolution process. Consequently, the linear-elastic assumption adopted in the macroscale regions is validated for the present loading scenario, as no damage is expected to occur in those regions. It should be acknowledged that, under different loading conditions or for structures with more distributed damage patterns, the damage zone could potentially extend beyond the initially designated mesoscale region. In such cases, the mesoscale subdomain would need to be enlarged accordingly. This is precisely the scenario that motivates the development of an adaptive refinement strategy, as discussed in Section 3.3, where the mesoscale region could be dynamically adjusted based on the evolving stress or damage field during the computation.

The damage pattern observed in the numerical simulation—tensile cracking initiating from the inner corner and propagating through the mortar matrix with a tortuous path deflected by aggregates—is qualitatively consistent with experimental observations of concrete failure under similar loading conditions. The dominance of the tensile failure mode and the crack trajectory through the weaker mortar phase, bypassing the stiffer aggregates, are well-documented characteristics of mesoscale concrete fracture. Quantitative comparison against specific experimental benchmarks, including load-displacement curves and crack patterns, would require careful calibration of CDP parameters to match a specific concrete mix and experimental setup, and is identified as an important direction for future work.

3.3. Discussions on Convergency and Accuracy

In order to apply the proposed nonlinear algorithm to engineering practices, the convergency and accuracy are investigated in this section. The influence of relaxation parameters and the overlapping area are also analyzed here.

According to Equation (23), error is estimated in the following way after each iteration step:

$$ϵ={\displaystyle \frac{{∥u^{n+1}-u^n∥}_2}{{∥u^{n+1}∥}_2}}$$

(25)

where ${∥·∥}_2$ is the $L^2$ norm of a vector, and $u^{n+1}$ are node displacements (vector) on the boundary of the overlapped macro-micro subdomains. Once the error is less than the given tolerance, i.e., $ϵ<{ϵ}_t$, the iterate finishes.

For comparison purposes, a benchmark model was developed by establishing the identical concrete structure with identical geometrical configuration, material constants, boundary conditions, and the identical element type CPE4R—the only difference is that the model is developed within one single scale and is meshed with a unified element size of 5 mm. The 5 mm mesh size was chosen as the benchmark because it represents a practical and commonly used mesh resolution for structural-level concrete analysis, consistent with the macroscale mesh employed in the multiscale model. The monotonic convergence trend confirms that the multiscale solution systematically approaches the benchmark as the tolerance decreases, and this fundamental convergence behavior is expected to be preserved regardless of the benchmark mesh refinement. Moreover, the mesoscale subdomain in the multiscale model already employs a finer mesh that captures the heterogeneous material structure in detail, which means that the critical region achieves a level of accuracy comparable to or even surpassing a uniformly fine single-scale model, but at a significantly lower computational cost.

To test the accuracy of results, four groups of nodes around the corner of the L-shape concrete structure are chosen, denoted as groups A, B, C, and D, as shown in Figure 4. The relative error of displacement on a specified node is calculated as

$$\eta ={\displaystyle \frac{u-u^{*}}{u^{*}}}$$

(26)

where u and $u^{*}$ are results from the multiscale model and the benchmark model, respectively, and $u^{*}$ is regarded as accurate result.

Figure 5 shows the relative error $\eta$ (Equation (26)) of displacement on each node versus the varying controlling tolerance ${ϵ}_t$. It could be seen that, as the tolerance ${ϵ}_t$ decreases, relative errors of the computed displacement on all nodes approach zero, indicating that the computed result could approach the accurate value uniformly.

The effect of a relaxation parameter on convergency is investigated. As seen in Equation (24), each subdomain could introduce a specific relaxation factor, denoted as ${\theta }_1$, ${\theta }_2$, and ${\theta }_3$, which correspond to the macroscale substructures A and B and the microscale substructure C, respectively.

First, the three relaxation factors are set to be the same, i.e., ${\theta }_1={\theta }_2={\theta }_3=\theta$. Results are illustrated in Figure 6, where the error is also calculated by Equation (25). It could be seen that, regardless of the choice of relaxation parameters, results can converge with the number of iterations increase, which further verify the convergency of the proposed algorithm for multiscale damage analysis.

A larger relaxation value can lift the convergence rate. However, as the number of iterations increases, the convergence rate decreases. Its accuracy may even become less than that with a smaller relaxation factor. As shown in Figure 6, the turning point for $\theta =0.3$ happens at the 150th–200th iteration compared with a smaller relaxation value and at the 70th–100th iteration compared with a larger relaxation value.

These results lead to a balanced suggestion on the choice of the relaxation parameter. If the error tolerance is relatively small, i.e., a higher accuracy is required, it is recommended to choose a smaller relaxation factor. In this case, it is appropriate to select between 0.2 and 0.3. On the other hand, if the accuracy requirement is relatively lower, hoping to exchange for a smaller number of iterations (a less time consumption), it is recommended to select a relatively larger relaxation factor, such as 0.4–0.6 in this case. It should be noted that these recommended ranges are empirical observations derived from the present problem configuration, and the specific optimal values may vary for different structural geometries, subdomain arrangements, and loading conditions. Nevertheless, the general trade-off between convergence rate and accuracy is consistent with the theoretical expectations from the classical overlapping domain decomposition literature [38], and the calibration strategy presented here provides a practical guideline for engineering applications.

Second, the situations of the three relaxation parameters that are not the same are investigated, as plotted in Figure 7. It can be seen from Figure 7a–c that only increasing the value of one certain factor has little effect on the convergence rate. When ${\theta }_1={\theta }_2$, an increase of ${\theta }_3$ has a slightly greater impact on the convergence. Similarly, this improvement effect only works in the circumstance of fewer iterations (here, fewer than about 100 iterations). The above results suggest a uniform choice of relaxation parameters on different subdomains, or better yet, a slightly higher factor on the microscale substructures than those on macroscale substructures, for nonlinear damage analysis is usually required on microscale domains. This finding has practical implications for the implementation of the framework: when computational resources are limited, assigning differentiated relaxation parameters that prioritize the convergence of the mesoscale subdomain can be a useful strategy to improve the overall efficiency of the iterative scheme without significantly compromising accuracy.

Taking the overlapped area in Figure 2 as the baseline, the overlapped size between domains A, C and domain B is increased and decreased by 50% by expanding and reducing the substructures A and C. The ±50% variation is selected to provide a representative range covering both notably reduced and substantially enlarged overlap configurations, enabling a clear observation of the trade-off between convergence rate and computational scale, as suggested by the classical ODD theory [38]. Results are graphically displayed in Figure 8.

For the present model, when the loop contains less than about 75 iterations, a larger overlap area leads to a higher accuracy. However, if the size of the iterate is beyond 150 steps or so, the accuracy curves tend to be flat, and in contrast, the smaller the overlap area, the higher the accuracy. It should also be taken into account that a larger overlap area may significantly increase the computational scale. In the condition of a controllable computational scale, the area of the overlap area could be appropriately increased, which may significantly promote the convergence rate.

Several additional remarks are made regarding the convergence, accuracy, and generality of the proposed framework.

Regarding the convergence criterion, the displacement-based norm (Equation (25)) is adopted because displacement is the primary unknown that is communicated and reconciled between subdomains in the ODD algorithm. As formulated in Algorithm 1, the iterative scheme exchanges displacement boundary conditions between overlapping subdomains, and convergence is therefore most naturally and directly measured by the consistency of displacement fields across the overlapping regions. This criterion is also universally applicable across both macroscale and mesoscale subdomains, providing a unified convergence measure for the entire multiscale system. In contrast, a damage-based criterion would only be meaningful within the mesoscale subdomain where damage evolution occurs, and would not capture the convergence of the macroscale elastic solution. Similarly, while an energy-based criterion could provide complementary information about the quality of the overall solution, it would require additional post-processing of stress and strain fields across all subdomains and introduce implementation complexity. The numerical results presented in Figure 5, Figure 6, Figure 7 and Figure 8 demonstrate that the displacement-based criterion effectively captures the convergence behavior of the algorithm. Nevertheless, incorporating energy-based or damage-based criteria as supplementary convergence measures could be a worthwhile refinement in future work, particularly for problems involving more complex damage patterns.

Regarding convergence behavior in the presence of high damage gradients, it is important to note that the additive Schwarz method converges successfully for all tested configurations, even though the damage variable reaches its limit of 1.0 in localized areas within the mesoscale subdomain. The monotonic error reduction observed in Figure 5, Figure 6, Figure 7 and Figure 8 confirms the overall convergence of the algorithm. However, the convergence rate does slow down as the number of iterations increases, particularly for larger relaxation parameters, which may be partially attributed to the strong nonlinearity introduced by damage localization and the associated high damage gradients. Several factors contribute to maintaining convergence under these challenging conditions: the pre-calculation strategy described earlier provides a physically meaningful initial guess that prevents divergence, the viscoplastic regularization in the CDP model smooths the damage evolution, and the domain decomposition framework itself confines the nonlinear behavior to the mesoscale subdomain so that the macroscale linear elastic solution remains stable throughout the iterations. For more severe loading scenarios or more complex damage patterns, additional convergence acceleration techniques such as preconditioning or coarse-grid correction may be beneficial.

Regarding the overlap geometry, only symmetric overlap configurations were tested in the present study, as this is the most natural arrangement for the L-shaped structure where the mesoscale subdomain is located at the intersection of two symmetric limbs. It should be emphasized that the additive Schwarz algorithm formulated in Algorithm 1 does not impose any restriction on the overlap geometry—the algorithm operates on general overlapping subdomains and is inherently capable of handling asymmetric, unidirectional, or irregular overlap configurations. Testing different overlap geometries for more complex structural configurations is a natural and important extension of this work, particularly when the critical damage zone is not centrally located or when the structural geometry does not permit symmetric decomposition.

Regarding the benchmark model, the 5 mm single-scale mesh was selected because it represents a practical and commonly used mesh resolution for structural-level concrete analysis with the CDP model. This mesh size is consistent with the macroscale mesh used in the multiscale model, ensuring a fair comparison under comparable computational conditions. A finer single-scale mesh would provide a more refined reference solution; however, the monotonic convergence trend observed in Figure 5 confirms that the multiscale solution systematically approaches the benchmark as the tolerance decreases, and this fundamental convergence behavior is expected to be preserved regardless of the benchmark mesh refinement. Moreover, the mesoscale subdomain in the multiscale model already employs a finer mesh that captures the heterogeneous material structure in detail, which means that the critical region achieves a level of accuracy comparable to or even surpassing a uniformly fine single-scale model, but at a significantly lower computational cost.

Regarding the computational cost and scalability of the proposed framework, the total computational time can be expressed as $T_{\mathrm{total}}=N_{\mathrm{iter}}\times [{max}_i(T_i)+T_{\mathrm{comm}}]$, where $N_{\mathrm{iter}}$ is the number of iterations required for convergence, $T_i$ is the wall-clock time for solving the i-th subdomain, and $T_{\mathrm{comm}}$ is the inter-subdomain communication time for interpolation and boundary condition updates. In a parallel computing environment, the subdomain computations are performed concurrently as specified in Algorithm 1, so the cost per iteration is governed by the most expensive subdomain rather than the sum of all subdomain times. Since the macroscale subdomains involve only linear elastic analysis with relatively coarse meshes, $T_i^{\mathrm{macro}}\ll T_i^{\mathrm{meso}}$, and the total time is dominated by the mesoscale subdomain computation. When scaling to larger structures with multiple mesoscale regions, the computational savings become more pronounced as the mesoscale regions represent a smaller fraction of the total structural domain, and the inherent parallelizability of the domain decomposition framework enables efficient distribution of subdomain computations across multiple processors.

Finally, regarding the generality of the relaxation parameter ranges, the suggested values (0.2–0.3 for higher accuracy requirements and 0.4–0.6 for faster convergence) are empirical observations derived from the present L-shaped concrete structure. These specific numerical values are influenced by the problem configuration, including the geometry, the number and arrangement of subdomains, the overlap size, and the degree of material nonlinearity. However, the general trends observed are consistent with the theoretical expectations from the classical ODD literature [38]: larger relaxation parameters tend to accelerate early convergence but may limit the ultimate accuracy, while smaller parameters provide better final accuracy at the cost of more iterations. The investigation of non-uniform relaxation parameters (Figure 7) further suggests that assigning a slightly higher factor to the microscale substructure than to the macroscale substructures can be beneficial, since the nonlinear damage analysis typically requires more iterative adjustment. When applying the framework to different structural configurations, similar parametric studies are recommended to identify the appropriate range, though the qualitative trends and the calibration strategy presented herein are expected to remain applicable.

4. Conclusions

Based on an overlapping domain decomposition method, a concurrent multiscale framework for damage analysis of concrete structures is formulated. The concurrent macro- and mesoscale regions are defined prior to the analyses. For regions where damage localization is expected to occur, a mesoscale model is employed to represent the concrete with mesoscopic features such as aggregates and mortar matrix. A macroscopic model with homogenized elastic parameters is employed for the regions where concrete material demonstrates elastic behaviors. The main conclusions are summarized as follows:

The proposed overlapping domain decomposition-based multiscale framework proves to be an efficient and practical tool for damage analysis of concrete structures. Concurrent computation for different regions is feasible in engineering practice.

The CDP model can be adopted to model the process of macroscopic crack formation, and thus is applicable to nonlinear problems in engineering structures.

Numerical results indicate that algorithm implementation in commercial software is feasible, and computational accuracy is related to the area of overlapping regions.

Last but not least, the proposed multiscale framework is briefly applied to a two-dimensional problem as a case in point. Nevertheless, when it comes to three-dimensional problems, computational cost will be tremendously increased. A more efficient algorithm will be expected, and this will be left for our forthcoming work. In future work, several extensions are envisioned, including an adaptive refinement strategy to dynamically adjust the mesoscale region based on evolving damage fields, exploration of asymmetric and irregular overlap geometries, and incorporation of supplementary energy-based or damage-based convergence criteria. The proposed framework is inherently material independent at the algorithmic level and can be adapted for different material types or specialized structural components (e.g., precast frames with replaceable plastic hinges, composite piers with ultra-high-performance concrete [45]) by replacing the CDP model with appropriate constitutive models. The framework is also fully compatible with stochastic aggregate generation procedures [18,42] and random field representations of material uncertainties, which would enable probabilistic predictions of structural response for reliability-based engineering design. From a practical perspective, the framework offers economic advantages by confining fine-scale analysis to critical regions, and its inherent parallelizability enables efficient utilization of modern computing resources. It should be noted that multiscale simulation results are intended to complement, rather than replace, code-based design procedures and experimental validation.