Parallelization Strategy for 3D Probabilistic Numerical Cracking Model Applied to Large Concrete Structures

Abstract

This work presents the application of a finite element model utilizing a three-dimensional (3D) probabilistic semi-explicit cracking model to analyze the rupture process of a large concrete wall beam. The numerical analysis predicts both the global behavior of the structure and its primary rupture mechanisms, utilizing three different finite element mesh refinements to ensure robustness. A Monte Carlo (MC) procedure is integrated into the modeling approach to account for probabilistic variations of the material properties. The statistical analysis derived from this probabilistic model may sometimes result in overly conservative safety coefficients, particularly when using a coarse mesh. Additionally, the detailed understanding of the structure’s cracking process, regardless of its rupture mechanism, may experience some reduction in precision. Due to the necessity of numerous simulations to achieve statistically significant results, the MC procedure can become computationally expensive. To address this, a straightforward parallelization of the Monte Carlo procedure was implemented, allowing multiple finite element analyses to be conducted concurrently. This strategy significantly reduced computational time, thereby enhancing the efficiency of the numerical model in performing numerical simulations of structural engineering.

1. Introduction

The use of computational parallelization strategies has become increasingly crucial in solving complex engineering problems, especially when a large number of equations need to be solved [1,2,3,4,5,6]. This is particularly true for the probabilistic models applied to the 3D simulation of concrete structures, which require numerous computational analyses due to the use of random distributions of material properties to represent heterogeneity [7,8,9]. In this context, computational parallelization facilitates more robust and detailed simulations by allowing the simultaneous execution of multiple analyses, significantly reducing processing time and increasing simulation efficiency.

Furthermore, the development of accurate numerical models for analyzing the rupture process in large concrete structures plays a pivotal role in ensuring their structural integrity and safety. A critical phenomenon that influences a structure’s response based on its size or volume, is the scale effect. This effect is attributed to concrete’s heterogeneous nature, physical and chemical phenomena during production, drying-induced micro-cracking, and water-induced porosity. Therefore, considering that the scale effect in the modeling of concrete structures is primordial due to the heterogeneity of the material, addressing it involves understanding both material-level constitutive relations and structure-level design methods and finite element analysis.

In this context, this work aims to analyze the cracking process in a very large concrete structure utilizing a three-dimensional (3D) probabilistic semi-explicit cracking model that has been developed and detailed in previous studies [8,10,11,12]. The model validation was conducted through the simulation of a fracture mechanical test using a large Double Cantilever Beam (DCB) specimen. The dimensions of this DCB specimen were significant, measuring 3.5 m in length, 1.1 m in width, and 0.3 m in height. Notably, the observed and modeled crack propagation length exceeded 2 m [13].

The structural case analyzed here is a concrete wall beam that can present more than one macrocrack unlike to the single macrocrack scenario observed in the DCB specimen [12], providing a more realistic representation of a structural behavior. To ensure the robustness of the model, three mesh refinements are employed to verify the numerical results independence of the finite element mesh.

The probabilistic numerical model focuses specifically on the propagation of macrocracks within concrete structures. To simulate the concrete heterogeneous behavior within the finite element method framework, it is necessary to use a Monte Carlo (MC) approach. Due to the need to use several samples to obtain statistically consistent results, the execution of the MC procedure becomes computationally very time-consuming, depending on the problem at hand. Therefore, given the computational cost of these simulations, optimizing the calculation time is critical.

As a solution to overcome this issue, a straightforward parallelization of the Monte Carlo procedure was implemented, allowing for multiple finite element analyses to be performed simultaneously. The study case demonstrates that the use of this strategy significantly reduces computational time, allowing the practical applicability of probabilistic numerical models in the analysis and design of real-world engineering problems of concrete structures.

The decision to enhance execution time by implementing the Monte Carlo (MC) method in parallel, rather than pursuing parallelization of both Finite Element Method (FEM) and MC codes, stems from some key factors. Firstly, the parallel implementation of the MC method offers simplicity in execution. Additionally, since FEM analyses are inherently independent tasks, there is no requirement for extensive communication overhead. This approach facilitates the concurrent execution of multiple FEM analyses on the computational platform, resulting in significant runtime reductions. Nonetheless, it is worth noting that future endeavors may explore enhancements to this strategy, including the investigation of dual-code parallelization involving both FEM and MC codes.

2. Principles of the Semi-Explicit Probabilistic Cracking Model

This study introduces a 3D semi-explicit probabilistic model for crack propagation, previously detailed in [12,14]. The model focuses specifically on analyzing crack propagation in concrete structures under mode I failures induced by tensile stresses. Developed using the FEM, the model incorporates considerations of heterogeneity and volume effects within a probabilistic framework, crucial for accurately simulating crack propagation in concrete structures. The model is implemented in FORTRAN [15,16] and integrated into a versatile FEM platform created by researchers from the Civil Engineering Program at COPPE/UFRJ. This platform has been widely applied across various studies, demonstrating its utility in diverse applications [1,8,9,17].

To address material characteristics such as heterogeneity and scale effects, the numerical model assumes that each finite element represents a distinct volume of heterogeneous material. This material’s behavior is influenced by its degree of heterogeneity $\left(r_e\right)$, and determined by the ratio of the finite element volume $\left(V_e\right)$ to the volume of the coarsest aggregate $\left(V_a\right)$:$(r_e=V_e/V_a)$. The model adopts a volume element-based approach to simulate macrocrack propagation within concrete structures. The criteria governing macrocrack propagation can be divided into two aspects: macrocrack initiation and macrocrack propagation. Macrocrack initiation depends directly on the uniaxial tensile strength $\left(f_t\right)$ of the concrete material. Once initiated, macrocrack propagation is guided by the dissipation of post-cracking energy in tension, ensuring the model accurately represents the mechanics of crack propagation in concrete structures. A macrocrack is considered fully propagated once all post-cracking dissipation energy is consumed.

An innovative feature of the model is its utilization of Linear Elastic Fracture Mechanics (LEFM) to derive post-cracking dissipation energy. This approach employs the mode I critical fracture energy $G_{IC}$ to quantify the energy dissipated during crack propagation. Both the uniaxial tensile strength and mode I critical fracture energy are treated as probabilistic mechanical characteristics within the model. The random distribution of $f_t$ follows the Weibull distribution [18,19], while $G_{IC}$ follows a lognormal distribution [20]. Accurately defining the parameters of these distributions is crucial for accurately representing material behavior.

At the microscale within each finite element (FE) volume, the cracking process results in energy dissipation governed by an isotropic damage law, as depicted in Equation (1), where the variables ${\tilde{\epsilon }}_0$, ${\tilde{\epsilon }}_{fi}$, and ${\tilde{\epsilon }}^k$, respectively, represent the damage initiation strain, maximum critical strain, and equivalent strain. This dissipative process initiates when the maximum principal stress (${\sigma }_1$) reaches the randomly assigned concrete tensile strength ($f_t$). Thereafter, once the total available energy for the FE is consumed, the element is considered cracked, and its stiffness matrix is considered null [11].

$$D=1-{\displaystyle \frac{{\tilde{\epsilon }}_0}{{\tilde{\epsilon }}^k}}\left[1-{\displaystyle \frac{({\tilde{\epsilon }}^k-{\tilde{\epsilon }}_0)}{({\tilde{\epsilon }}_{fi}-{\tilde{\epsilon }}_0)}}\right]$$

(1)

An illustration of the main principles of the proposed model is depicted in Figure 1. This figure highlights the random distribution of concrete’s tensile strength and the volumetric density of dissipated energy, represented as $g_{IC}$, along with the elementary representation of the damage evolution process. Each finite element follows unique laws governing this process due to inherent randomness. The determination of $g_{IC}$ utilizes an energetic regularization technique [21], which relates the material fracture energy $G_{IC}$ to the elementary characteristic length $l_e$ through the equation: $g_{IC}=G_{IC}/l_e$, where $l_e={\left(V_e\right)}^{1/3}$.

In this modeling approach, it is important to emphasize cracks emerge and propagate at a macroscopic scale due to the elemental failure of successive elements, which occur randomly and may coalesce to form macroscopic cracks. A finite element (FE) is considered cracked only when its associated energy is fully dissipated. Therefore, within this framework, the model does not address traditional crack propagation laws as seen in classical damage model or smeared crack model [22,23,24,25,26,27,28], as the inclusion of the damage parameter in the constitutive model solely serves to dissipate $G_{IC}$.

Moreover, in this approach, each finite element analysis acts as a sample within the Monte Carlo procedure, yielding global structural responses of the model. This methodology involves executing multiple FEM simulations for a given structural scenario, incorporating diverse spatial distributions of mechanical material properties to ensure statistically significant outcomes.

2.1. Model’s Parameters Estimation

Accurately determining parameters for the Weibull and lognormal distributions is crucial to ensure the model’s consistency. The Weibull distribution is characterized by shape and scale parameters $(b,c)$, while the lognormal distribution is defined by mean and standard deviation parameters $(\mu ,\sigma )$. These parameters govern the stochastic distribution of tensile strength and fracture energy, capturing the heterogeneous behavior of materials at the finite element scale.

Fracture energy ($G_{IC}$) is assumed to be an intrinsic property of the material with a constant mean value $\mu \left(G_{IC}\right)$, independent of scale and specific to the type of concrete. The standard deviation $\sigma \left(G_{IC}\right)$ of the lognormal distribution thus becomes a key parameter to be determined. This assumption aligns with findings from Rossi [13], acknowledging inherent material variability. Therefore, the primary parameters to be determined are as follows: $(b,c)$ for the Weibull distribution and $\sigma \left(G_{IC}\right)$ for the lognormal distribution. Recent advancements [12,14] now enable the estimation of these model parameters using concrete compressive strength, maximum aggregate volume, and finite element mesh size data. This capability significantly enhances the model’s versatility and applicability across diverse concrete compositions.

The dependency of $\sigma \left(G_{IC}\right)$ on finite element volume was a significant outcome in the model’s development [8,12]. This relationship was established through inverse analysis of a concrete mix design, validated by simulating a macrocrack propagation test on a large DCB specimen and comparing results with experimental data from Rossi [13]. An equation was derived correlating the lognormal distribution’s standard deviation with mesh heterogeneity. Subsequently, the application of the model was extended to include other concrete mixes with compressive strengths below 130 MPa and aggregate diameters of 10 mm to 20 mm, by proposing a methodology to estimate the parameters of $G_{IC}$ [14].

For the Weibull distribution parameters, an iterative numerical method was developed. This method integrates expressions for the mean tensile strength $\mu \left(f_t\left(r_e\right)\right)$ and coefficient of variation ${\displaystyle \frac{\sigma }{\mu }}\left(f_t\left(r_e\right)\right)$, derived from experimental data and scaling laws. It estimates $(b,c)$ parameters for specific concrete volumes, validated for compressive strengths up to 130 MPa and maximum aggregate sizes of 10 mm or more. Applied directly at the finite element level, this method assigns $(b,c)$ parameters to each element based on its volume, maximum aggregate size, and compressive strength ($f_c$).

2.2. Parallelization of the Monte Carlo Method

In this study, a Monte Carlo procedure is used for simulating probabilistic concrete cracking, introducing stochasticity at the material’s local scale. Each MC sample corresponds to a finite element problem solution, treated as an independent sample. Random variables are generated for each sample based on the probability density functions, yielding load-displacement diagrams $(P$-$\delta )$ as the outcome.

The computational demands of the MC procedure increase notably due to each sample involving solving a finite element problem. To address this, a parallel programming strategy was employed to implement the MC method using the OpenMP application programming interface. OpenMP facilitates the development of multi-threaded applications in languages such as C/C++ and FORTRAN. Comprising compilation directives, function libraries, and environment variables, OpenMP supports both explicit and implicit multi-threaded parallelism, making it a versatile tool for parallel computing [29,30,31].

The parallel algorithm for the Monte Carlo method employs a master-slave parallel model, which is widely utilized in networked computing environments [32]. In this approach, the code initiates with a master thread that executes sequentially until entering a parallel region. During this phase, the master thread spawns multiple parallel threads, each executing tasks concurrently within the parallel region. Upon the completion of tasks, the threads synchronize and terminate, leaving only the master thread. This parallel model aligns with the sequential algorithm’s principles, facilitating the simultaneous execution of multiple independent FEM analyses. The main steps of the Parallel Monte Carlo (PMC) procedure are outlined in Algorithm 1. This method was selected for its straightforward implementation and compatibility with the stochastic nature of the proposed methodology, which necessitates a large number of independent simulations.

Algorithm 1 Parallel Monte Carlo Procedure

1: Variables initialization  2: Read maxmc and stop criteria ; // maxmc is the maximum number of samples  3: Read nthreads ; // nthreads is the number of threads  4: j = 0 ; // counter of performed samples  5: while stopping criterion is not reached do  6:     j = j + nthreads  7:     Initialize the parallel environment  8:            For each block with nthreads samples do  9:                   Perform the FEM analysis 10:                  Save $(P,\delta )$ curve and relevant results of the sample 11:            end for 12: Finalize the parallel environment 13: end while 14: Perform the statistical analysis

The selected stopping criterion for PMC simulations was reaching the predefined number of samples. Because FEM analyses differ due to the problem’s probabilistic nature, tasks are unevenly balanced. Thus, a dynamic schedule clause was utilized in PMC implementation, distributing tasks based on execution time, with new tasks assigned as threads become idle.

3. Structural Case and Numerical Simulations

The structural case investigated in this study involves a sizable wall beam subjected to four-point bending, characterized by dimensions of 5 m in length, 3 m in height, and 0.5 m in thickness. A representation of the test configuration is shown in Figure 2, where P is the total force applied to the specimen by two loading pins; d is the distance between the supporting and loading pins; l is the specimen width, and h is the specimen height. The distance between the bottom loading supports is 5 m and between the upper loading supports is 2.2 m.

In contrast to common practices involving large reinforced concrete beams, often with or without fibers, the beam used as the structural case in this study comprises plain concrete without reinforcements. The primary objective of the numerical simulations is to demonstrate the model’s applicability. Through comparisons across different levels of mesh refinement, the study aims to highlight the model’s robustness and independence from mesh density. Additionally, the investigation seeks to validate the efficacy of the estimated probabilistic parameters.

In the numerical simulations, three-dimensional finite element analyses were conducted using three distinct mesh refinements. The refinement level of each mesh is determined by the parameter $r_e^{mean}$, calculated as $V_e^{mean}/V_a$, where $V_e^{mean}$ represents the mean volume of the finite elements within each mesh. The meshes consist of tetrahedral elements approximated by linear interpolation functions. Visual representations of the three finite element meshes can be found in Figure 3, Figure 4 and Figure 5, while detailed information regarding the meshes, including the number of elements and nodes, as well as the maximum and minimum volumes of the finite elements, is provided in

Table 1. Description of the three finite element meshes.

Reference	Neq	Nnode	Numel	${\mathit{r}}_{\mathit{e}}^{\mathbf{mean}}$	${\mathit{r}}_{\mathit{e}}^{\mathbf{sd}}$	${\mathit{r}}_{\mathit{e}}^{\mathbf{min}}$	${\mathit{r}}_{\mathit{e}}^{\mathbf{max}}$
Mesh 1	7180	2403	10,364	829.1	276.3	110.6	2963.9
Mesh 2	22,756	7595	37,503	223.3	63.1	54.4	710.5
Mesh 3	64,243	21,424	112,572	73.9	22.3	14.4	207.4

.

The mechanical characteristics of concrete, which are identical to those of the DCB previous modeling [8,14], are the following: compressive strength $f_c$ = 54 MPa; splitting tensile strength $f_t$ = 4.1 MPa; Young modulus E = 35.5 MPa; fracture energy $G_{IC}$ = 1.25 $\times {10}^{-4}$ MN/m. The maximum aggregate size is 12 mm, corresponding to $V_a=0.9$ cm³. Across all simulations, a total displacement of 2.5 mm is applied, divided into 50 steps of 0.05 mm each. Additionally, thirty samples are analyzed for each Monte Carlo simulation.

3.1. Numerical Results

All results are presented in terms of the global behavior of the structure, specifically load-displacement curves depicting the wall beam’s overall mechanical response. The Monte Carlo simulation results for the three mesh refinements are displayed in Figure 6, Figure 7 and Figure 8, encompassing 30 finite element analyses in each case. Figure 9 shows a comparison between the arithmetic mean curves obtained for each mesh refinement. As can be observed, the load-displacement curves for the three meshes exhibit similar behavior, particularly in the cases of Mesh 2 and Mesh 3, indicating coherence in determining the statistical parameters.

3.2. Analysis of the Results

Upon examining Figure 6, Figure 7 and Figure 8, it becomes evident that the dispersion associated with the global behavior of the structure decreases as the mesh is refined. Statistically, this suggests that using Mesh 1 (the coarsest option) would yield a larger safety coefficient compared to Mesh 2 and especially Mesh 3.

From the analysis of the results presented in Figure 9 the following can be stated: (1) All three meshes exhibit a similar global behavior for the structure. This brittleness aligns well with the nature of the considered structural problem, given that the structure lacks reinforcement; (2) The mesh dependency remains minimal, even with a substantial difference in refinements between Meshes 1 and 3 (there is practically no difference in behavior between meshes 2 and 3).

Examining Figure 10, Figure 11 and Figure 12 it can be observed that the primary crack mechanism leading to the structural rupture remains consistent across all three meshes, manifesting as a macrocrack originating from the left bottom loading support and extending to the left upper loading support. However, the detailed description of the cracking pattern diverges between Mesh 1 and Meshes 2 and 3. With Mesh 1, only a single macrocrack is observed. In contrast, Meshes 2 and 3 exhibit a more intricate pattern, featuring a second macrocrack initiating from the right bottom loading support.

In summary, it can be affirmed that the refinement of Mesh 1 is sufficient to yield an acceptable global behavior of the structure, possibly with an overly conservative safety coefficient. It also facilitates a proper rupture mechanism. Nevertheless, it does not attain the level of fineness required for a detailed analysis of the cracking process.

4. Parallelization Performance Analysis

A parallel processing environment typically consists of multiple processors with shared or distributed memory, an operating system for managing processes, and a parallel algorithm [33]. In distributed memory systems, each processor has its local memory, and data is exchanged via a high-speed network using a message system like MPI (Message Passing Interface). Shared memory systems feature multiprocessors that share a single memory space, communicating through shared variables. Hybrid systems combine both forms, with nodes that have separate address spaces and multiple processors sharing memory within each node [34].

In this study, performance simulations are conducted on a parallel computing platform consisting of a cluster comprising four nodes interconnected by a local Ethernet network with a speed of 10 Gb/s. Each node is equipped with dual processors utilizing a cache-coherent Non-Uniform Memory Access (cc-NUMA) architecture [35]. Detailed specifications of the hardware for each node can be found in

Table 2. General characteristics of each node of the cluster.

Characteristic	Description
Processor	Intel Xeon E5-2630 v4
Number of processors	2
Cores	2 × 10
Socket per node	2
Number of threads	20
Cache size (L3)	25 MB
Clock	2.2 Ghz
Memory per node	256 GRAM (DDR4-2134 MHz)
Motherboard	Intel S2600CWR
LAN	Gigabit (2 × 10 Gbit)
Operational System	CentOS 7

. Communication between nodes on this platform occurs via the local network, while intra-node communication between processors is managed by a Quick Path Interconnect (QPI) circuit [3].

There are several metrics available to gauge the performance of parallel applications, the most common being speedup, efficiency, and FLOPs (floating-point operations per second). In this work, speedup is selected as the primary metric due to its practicality and clarity in demonstrating how much faster the parallel code runs compared to its sequential counterpart. Speedup (denoted as $S_n)$ is defined as the ratio of the sequential execution time $T_s$ to the parallel execution time $T_n$. Mathematically, $S_n$ is calculated as:

$$S_n={\displaystyle \frac{T_s}{T_n}}$$

(2)

It is noteworthy that the execution times reported in this work refer to the wall time, which is the actual elapsed time of the program’s execution. This time measurement is preferable to CPU time for evaluating parallel performance, and therefore, it is adopted in this work.

4.1. Description of the Performance Analysis

The parallelization strategy was designed with the understanding that each MC sample corresponds to an independent FEM analysis, while the probabilistic model response relies on multiple analyses of the problem. Hence, a direct parallelization approach was adopted, enabling the concurrent execution of multiple finite element analyses. During performance tests, simulations of parallel Monte Carlo were conducted with each Monte Carlo simulation (set of samples) running on a separate node of the cluster, thereby constituting shared memory tests.

The number of samples in each MC simulation during performance tests is set to twenty ($n_{\mathrm{mc}}=20$) to align with the available threads per cluster node, optimizing speedup test duration. Consequently, the first twenty FE analyses per problem solution are used for each case. Consistency is ensured by using saved random seeds for initializing random property generation, thereby maintaining identical scenarios for precise runtime comparisons.

The parallel computational platform, consisting of four nodes, enabled the simultaneous execution of up to four problems throughout the performance analysis phase. This capability provided a significant advantage for executing the inverse analysis procedure to estimate model parameters and evaluate speedup tests. However, since these cases were independently performed on each cluster node utilizing available hardware resources, a specific study quantifying the time saved by concurrently using all four nodes was not conducted, despite the intuitive notion of time savings.

4.2. Performance Results

The results of parallelizing the Monte Carlo procedure are illustrated in Figure 13, Figure 14, Figure 15 and Figure 16. Figure 13 provides a comparison of the speedup obtained for each mesh refinement, while Figure 14, Figure 15 and Figure 16 show (a) the speedup measures, and (b) the execution time. The speedup is measured relative to the number of threads ($nt$) used, with $nt$ taking the values of 1, 2, 5, 10, and 20.

The results indicate that Mesh 1 exhibits superior speedup compared to Meshes 2 and 3. Meshes 2 and 3 demonstrate comparable outcomes, though consistently inferior to Mesh 1. The overall reduction in execution time, considering both sequential and 20-thread runtimes, is as follows for the three cases: Mesh 1 decreases from 1.16 h to 0.08 h; Mesh 2 decreases from 5.48 h to 0.46 h and Mesh 3 decreases from 25.65 h to 2.2 h.

Discussions and Remarks

Based on the reported results, it is evident that the PMC has notably decreased the execution time of the simulated cases. Concerning the speedup measures, it can be seen that as the number of threads increases, the speedup values start to deviate from the ideal speedup curve. This difference becomes more pronounced for Mesh 1 when 20 threads are used (15.27) whereas, for Meshes 2 and 3, it occurs as early as 10 threads (7.86 and 7.52, respectively). The values of speedup achieved with 20 threads for Meshes 2 and 3 are similar, 11.99 and 11.66, respectively. It is important to mention that since each sample’s runtime varies depending on the distribution of elementary random parameters, a loss of efficiency may occur when there is a significant difference between the runtimes of different samples, despite using dynamic scheduling in the parallel region implementation.

To assess the impact of varying sample execution times on parallelization performance, additional Monte Carlo (MC) simulations were conducted. These simulations focused on an illustrative scenario where all MC samples are identical, sharing the same elementary properties. The aim of these hypothetical cases is to determine the maximum potential gain achievable through parallelization. The results of the speedups obtained for these new MC simulations are presented in

Table 3. Comparison between the speedups obtained for the real cases and the hypothetical cases.

Mesh Reference	nt	Distinct Samples	Equal Samples
Mesh 1	1	1.00	1.00
	2	1.97	2.05
	5	4.76	5.10
	10	9.23	10.33
	20	15.27	19.86
Mesh 2	1	1.00	1.00
	2	1.98	2.02
	5	4.44	5.04
	10	7.86	9.94
	20	11.99	16.29
Mesh 3	1	1.00	1.00
	2	1.93	1.96
	5	4.22	4.65
	10	7.52	8.93
	20	11.66	13.87

, where $nt$ represents the number of threads used. The results are categorized as “distinct samples”, reflecting those previously reported in Section 4.2, and “equal samples”, representing the hypothetical scenario described here.

As can be observed, the use of identical samples enhances parallelization performance across all analyzed scenarios. The performance improvement percentages for the three mesh refinements, when utilizing 20 threads, are 30.04%, 35.83%, and 18.92%, respectively. For this illustrative scenario, the best speedup is obtained for Mesh 1 (19.86) and it nearly reaches the ideal speedup value when 20 threads are used. For Mesh 2, the performance gain is closer to the ideal speedup up to the use of 10 threads, with a deviation of the ideal value when 20 threads are used. Conversely, for Mesh 3, the speedup values start to deviate from the ideal speedup from the use of 10 threads.

As evident, employing identical samples significantly enhances parallelization performance across all analyzed mesh refinements. The performance improvement percentages for the three mesh refinements when utilizing 20 threads are 30.04%, 35.83%, and 18.92%, respectively. In this illustrative scenario, the most substantial speedup achieved with 20 threads is for Mesh 1 (19.86), nearly reaching the ideal speedup. Additionally, a super-linear speedup is observed when 10 threads are used (10.33). For Mesh 2, the results deviate from the ideal speedup with 20 threads (16.29), yet exhibit a slight super-linear speedup with five threads (5.04) and approximate the ideal with 10 threads (9.94). Conversely, for Mesh 3, speedup values begin deviating from the ideal with 10 threads (8.93), with a more pronounced impact when 20 threads are employed (13.87). In all cases, it is observed that a speed-down when all cores are used. This behavior may stem from cache memory access saturation due to the utilization of all cores [3].

5. Conclusions

In summary, this study applies a 3D probabilistic semi-explicit cracking model to analyze macrocrack propagation in large structures using a numerical simulation of a wall beam as a case study. This structural scenario illustrates situations where more than one macrocrack can develop. The results demonstrated mesh independence and underscored the model’s reliability in predicting both global behavior and primary rupture mechanisms within structures. While the model showcases robustness, it is crucial to exercise caution with coarser meshes, as they may yield conservative safety coefficients and reduce the precision of the detailed cracking process, irrespective of rupture mechanisms.

The incorporation of parallel Monte Carlo into the probabilistic approach has significantly reduced simulation times across diverse mesh configurations. Despite its simplicity, this integration represents a substantial advancement within the framework of this probabilistic approach. Notably, for Mesh 1, runtime decreased from 1.16 h to 0.08 h; for Mesh 2, from 5.48 h to 0.46 h and for Mesh 3, from 25.65 h to 2.2 h. This advancement is crucial for facilitating extensive simulations required by this modeling approach, marking a significant stride toward applying the probabilistic model effectively in three-dimensional analyses for large-scale structures.

Looking ahead, further refinement of the model is essential for practical applications in real-world large concrete constructions, particularly in modeling steel rebars and interactions between concrete and steel. Additionally, the model should be expanded to simulate macrocrack propagation in fiber-reinforced concrete, thereby broadening its applicability in structural analysis and design to encompass more comprehensive scenarios.