Laplacian Smoothing Research Articles

Benchmark simulation of the flow-induced vibrations for nuclear applications

This study investigates the computational modelling of flow-induced vibrations of cantilever rods subjected to turbulent axial flow at operating conditions relevant to those of fuel rods of pressurized-water-cooled (PWR) nuclear reactors. The aim is to assemble all the modelling elements needed for a cost-effective and thus URANS-based (Unsteady Reynolds Averaged Navier–Stokes) modelling strategy, employing high-Reynolds-number models of turbulence. This objective is pursued through three stages. First, we investigate the numerical FSI (Fluid–Structure Interaction) strategy adopted, through the computation of flow-induced vibration of an elastic plate subjected to axial laminar flow. We subsequently assess the suitability of URANS models through computations of turbulent flow over a forward-facing step, for which measurements of the fluctuating wall pressure are available. On the numerical side, these explorations led to adopting a two-way FSI strategy, using a single finite-volume solver, with the Arbitrary Lagrangian–Eulerian (ALE) approach, high order convective discretization schemes and Laplacian smoothing for the displacement of the mesh in the fluid domain. On the physical modelling side, they resulted in the use of high-Reynolds-number Reynolds stress transport models. The resulting modelling strategy is subsequently validated against the experimentally investigated case of a steel cantilever rod caused to vibrate through exposure to turbulent axial flow. The resulting comparisons show that for the first time, to our knowledge, both the frequency and the amplitude of the flow induced vibrations of this case, have been reproduced with good accuracy.

Investigation on the smoothing algorithms used in the coarse aggregate texture characterization

In order to characterize the three-dimensional texture of coarse aggregates, the Laplacian, distance-weighted Laplacian, cotangent-weighted Laplacian and Taubin smoothing algorithms are considered in this study. Two smoothing scale parameters are added in the algorithms to control the smoothing operation within the texture level. Two standard geometric models (sphere and cube) are constructed to compare the effectiveness of the four smoothing algorithms in removing the surface texture. The results show that the Taubin smoothing algorithm has an obvious advantage over the other three algorithms. The three parameters (smoothness distance, smoothness level and negative scaling factor) in the Taubin smoothing algorithm are further investigated. Based on the study, the smoothness distance is suggested to be between one time the size of mesh and 2 times the size of mesh and should be determined according to the parameters analysis results. The smoothness level is suggested to be 1.1 and the negative scaling factor is suggested to be −1.5. The advantage of the two smoothing scale parameters is also studied. The results show that the two smoothing scale parameters can effectively control the smoothing operation within the given texture level. Finally, the effectiveness of the Taubin smoothing algorithm in coarse aggregate texture characterization is verified by the actual coarse aggregate models. The results demonstrate that the Taubin smoothing algorithm can effectively distinguish the texture of coarse aggregates and can be used to characterize the coarse aggregate texture.

Parameterized implementation of a triple refined modeling framework for mesoscale concrete

• A refined modeling framework for mesoscale concrete is developed using the parametric method. • The refined geometric aggregate models that consider micro/meso-scale morphology are generated. • The refined aggregate placement process is performed using the 3D BPT and Bullet physics engine. • The refined material assignment scheme is applied to describe random-continuous fields of material properties. As a typical heterogeneous composite material, concrete’s complex physical and mechanical behaviors on the macroscale are controlled by its mesoscale components and material properties. This paper aims to develop a refined modeling framework for mesoscale concrete using the parametric method, including the refined geometric aggregate model, refined aggregate placement process, and refined material assignment scheme. The geometric aggregates that consider micro/meso-scale morphology are first generated procedurally. Different degrees of mesh decimating operations are applied to the aggregates, and then the aggregate surface is smoothed by the Laplacian smoothing algorithm to build aggregate libraries of different resolutions. The numerical framework proposed by Huang et al. [39] is adopted in this work to conduct 3D bin packing and simulate the compaction and vibration process through the Bullet physics engine, so as to obtain the mesoscale geometric model with a high aggregate content. Furthermore, the material assignment schemes based on the statistical distribution and spatial correlation theory are applied to describe random-continuous fields of material properties. The chloride diffusion coefficient is used to illustrate the material assignment process. The triple refined modeling framework proposed in this paper provides a reference for high-fidelity parametric modeling of mesoscale concrete.

LRTCLS: low-rank tensor completion with Laplacian smoothing regularization for unveiling the post-transcriptional machinery of N6-methylation (m6A)-mediated diseases.

Recently, N6-methylation (m6A) has recently become a hot topic due to its key role in disease pathogenesis. Identifying disease-related m6A sites aids in the understanding of the molecular mechanisms and biosynthetic pathways underlying m6A-mediated diseases. Existing methods treat it primarily as a binary classification issue, focusing solely on whether an m6A-disease association exists or not. Although they achieved good results, they all shared one common flaw: they ignored the post-transcriptional regulation events during disease pathogenesis, which makes biological interpretation unsatisfactory. Thus, accurate and explainable computational models are required to unveil the post-transcriptional regulation mechanisms of disease pathogenesis mediated by m6A modification, rather than simply inferring whether the m6A sites cause disease or not. Emerging laboratory experiments have revealed the interactions between m6A and other post-transcriptional regulation events, such as circular RNA (circRNA) targeting, microRNA (miRNA) targeting, RNA-binding protein binding and alternative splicing events, etc., present a diverse landscape during tumorigenesis. Based on these findings, we proposed a low-rank tensor completion-based method to infer disease-related m6A sites from a biological standpoint, which can further aid in specifying the post-transcriptional machinery of disease pathogenesis. It is so exciting that our biological analysis results show that Coronavirus disease 2019 may play a role in an m6A- and miRNA-dependent manner in inducing non-small cell lung cancer.

Laplacian smoothing gradient descent

We propose a class of very simple modifications of gradient descent and stochastic gradient descent leveraging Laplacian smoothing. We show that when applied to a large variety of machine learning problems, ranging from logistic regression to deep neural nets, the proposed surrogates can dramatically reduce the variance, allow to take a larger step size, and improve the generalization accuracy. The methods only involve multiplying the usual (stochastic) gradient by the inverse of a positive definitive matrix (which can be computed efficiently by FFT) with a low condition number coming from a one-dimensional discrete Laplacian or its high-order generalizations. Given any vector, e.g., gradient vector, Laplacian smoothing preserves the mean and increases the smallest component and decreases the largest component. Moreover, we show that optimization algorithms with these surrogates converge uniformly in the discrete Sobolev $$H_\sigma ^p$$ sense and reduce the optimality gap for convex optimization problems. The code is available at: https://github.com/BaoWangMath/LaplacianSmoothing-GradientDescent .

Effective Deep Attributed Network Representation Learning With Topology Adapted Smoothing.

Attributed networks are ubiquitous in the real world, such as social networks. Therefore, many researchers take the node attributes into consideration in the network representation learning to improve the downstream task performance. In this article, we mainly focus on an untouched "oversmoothing" problem in the research of the attributed network representation learning. Although the Laplacian smoothing has been applied by the state-of-the-art works to learn a more robust node representation, these works cannot adapt to the topological characteristics of different networks, thereby causing the new oversmoothing problem and reducing the performance on some networks. In contrast, we adopt a smoothing parameter that is evaluated from the topological characteristics of a specified network, such as small worldness or node convergency and, thus, can smooth the nodes' attribute and structure information adaptively and derive both robust and distinguishable node features for different networks. Moreover, we develop an integrated autoencoder to learn the node representation by reconstructing the combination of the smoothed structure and attribute information. By observation of extensive experiments, our approach can preserve the intrinsical information of networks more effectively than the state-of-the-art works on a number of benchmark datasets with very different topological characteristics.

Biharmonic 3D Parametric Human Model

3D human body parametric model is a kind of high-level semantic information, which can provide effective prior knowledge for 3D human body reconstruction. The existing human body parametric model does not consider the local similarity characteristics of parameters, and it is difficult to guarantee the smooth model. A 3D human body parametric model based on the BlendSCAPE model is proposed. The biharmonic constraint is considered in every deformation step of the model. Then, each parameter is trained in a data-driven way, and the whole training process is redesigned with the introduction of biharmonic constraint. The training of parameters is finished by solving some minimum energy functions. Finally, experiments are carried out on CAESAR datasets. The results show that the model has a 14.2% reduction in the real data fitting error compared with BlendSCAPE model. Through generating human shapes with different poses and shapes and comparing the mesh before and after Laplace smoothing, the smoothness of the model is improved by 7.3%. The pose estimation results of real data in A-pose show that the deviation of the body part of the model is reduced by 18.6%.

Self-Supervised Graph Neural Networks via Diverse and Interactive Message Passing

By interpreting Graph Neural Networks (GNNs) as the message passing from the spatial perspective, their success is attributed to Laplacian smoothing. However, it also leads to serious over-smoothing issue by stacking many layers. Recently, many efforts have been paid to overcome this issue in semi-supervised learning. Unfortunately, it is more serious in unsupervised node representation learning task due to the lack of supervision information. Thus, most of the unsupervised or self-supervised GNNs often employ \textit{one-layer GCN} as the encoder. Essentially, the over-smoothing issue is caused by the over-simplification of the existing message passing, which possesses two intrinsic limits: blind message and uniform passing. In this paper, a novel Diverse and Interactive Message Passing (DIMP) is proposed for self-supervised learning by overcoming these limits. Firstly, to prevent the message from blindness and make it interactive between two connected nodes, the message is determined by both the two connected nodes instead of the attributes of one node. Secondly, to prevent the passing from uniformness and make it diverse over different attribute channels, different propagation weights are assigned to different elements in the message. To this end, a natural implementation of the message in DIMP is the element-wise product of the representations of two connected nodes. From the perspective of numerical optimization, the proposed DIMP is equivalent to performing an overlapping community detection via expectation-maximization (EM). Both the objective function of the community detection and the convergence of EM algorithm guarantee that DMIP can prevent from over-smoothing issue. Extensive evaluations on node-level and graph-level tasks demonstrate the superiority of DIMP on improving performance and overcoming over-smoothing issue.

Comparability of skeletal fibulae surfaces generated by different source scanning (dual-energy CT scan vs. high resolution laser scanning) and 3D geometric morphometric validation.

This work aims to test accuracy and comparability of 3D models of human skeletal fibulae generated by clinical CT and laser scanner virtual acquisitions. Mesh topology, segmentation and smoothing protocols were tested to assess variation among meshes generated with different scanning methods and procedures, and to evaluate meshes‐interchangeability in 3D geometric morphometric analysis. A sample of 13 left human fibulae were scanned separately with Revolution Discovery CT dual energy (0.625 mm resolution) and ARTEC Space Spider 3D structured light laser scanner (0.1 mm resolution). Different segmentation methods, including half‐maximum height (HMH) and MIA‐clustering protocols, were compared to their high‐resolution standard generated with laser‐scanner by calculating topological surface deviations. Different smoothing algorithms were also evaluated, such as Laplacian and Taubin smoothing. A total of 142 semilandmarks were used to capture the shape of both proximal and distal fibular epiphyses. After Generalized Procrustes superimposition, the Procrustes coordinates of the proximal and distal fibular epiphyses were used separately to assess variation due to scanning methods and the operator error. Smoothing algorithms at low iteration do not provide significant variation among reconstructions, but segmentation protocol may influence final mesh quality (0.09–0.24 mm). Mean deviation among CT‐generated meshes that were segmented with MIA‐clustering protocol, and laser scanner‐generated ones, is optimal (0.42 mm, ranging 0.35–0.56 mm). Principal component analysis reveals that homologous samples scanned with the two methods cluster together for both the proximal and distal fibular epiphyses. Similarly, Procrustes ANOVA reveals no shape differences between scanning methods and replicates, and only 1.38–1.43% of shape variation is due to scanning device. Topological similarities support the comparability of CT‐ and laser scanner‐generated meshes and validate its simultaneous use in shape analysis with potential clinical relevance. We precautionarily suggest that dedicated trials should be performed in each study when merging different data sources prior to analyses.

A novel dynamic adaptive unstructured mesh algorithm for simulating multi‐object relative motion in incompressible fluid

AbstractA novel dynamic adaptive unstructured mesh (DAUM) algorithm is proposed to solve incompressible multi‐object relative motion (MORM). The DAUM algorithm, consisting of exponential function deformation, adaptive edge swapping, and area Laplace smoothing, can greatly improve dynamic mesh robustness and perfectly overcome mesh skewness. The core of DAUM is the adaptive edge swapping inspired by Delaunay triangulation, which is distinguished from traditional edge swapping. The adaptive edge swapping can fully consider the relationship of neighbor elements only using Delaunay triangulation. Meanwhile, the implementation and reliability of adaptive edge swapping are better than the traditional edge swapping method due to eliminating the interference of the non‐convex polygon, so more code remedies can be avoided. Using the DAUM, none of the vertices is inserted or deleted so that the manipulation of the dynamic mesh is easily implemented and maintains computational efficiency in the process of mesh motion. Three representative geometries are used to assess the performance of the DAUM in MORM. To systematically analyze the advantages of DAUM, two relatively moving cylinders have been numerically investigated in incompressible flow. The inline force and lift force profiles on two cylinders are obtained and analyzed by using the flow field information. Three interaction stages are divided based on the parameter G and the interactional intensity of two inner anticlockwise vortices is considered as the division criteria. At the running process of the DAUM algorithm, the dynamic mesh quality is well controlled and remains in the high‐quality range based on the aspect ratio (AR) criterion. The results indicate that the proposed DAUM algorithm can properly solve the difficulties caused by MORM, especially for period oscillation motion.

Smoothing the Undersampled Carpal Bone Model with Small Volume and Large Curvature: A Feasibility Study.

The carpal bones are eight small bones with irregularities and high curvature on their surfaces. The 3D model of the carpal bone serves as the foundation of further clinical applications, e.g., wrist kinematic behavior. However, due to the limitation of the Magnetic Resonance Imaging (MRI) technique, reconstructed carpal bone models are discretely undersampled, which has dramatic stair-step effects and leads to abnormal meshes on edges or surfaces, etc. Our study focuses on determining the viability of various smoothing techniques for a carpal model reconstructed by in vivo gathered MR images. Five algorithms, namely the Laplacian smoothing algorithm, the Laplacian smoothing algorithm with pre-dilation, the scale-dependent Laplacian algorithm, the curvature flow algorithm, and the inverse distance algorithm, were chosen for evaluation. The assessment took into account the Relative Volume Difference and the Hausdorff Distance as well as the surface quality and the preservation of morphological and morphometric properties. For the five algorithms, we analyzed the Relative Volume Difference and the Hausdorff Distance for all eight carpal bones. Among all the algorithms, the scale-dependent Laplacian method processed the best result regarding surface quality and the preservation of morphological and morphometric properties. Based on our extensive examinations, the scale-dependent Laplacian algorithm is suitable for the undersampled carpal bone model with small volume and large curvature.

Analysis of Inflation Rate by using a Discrete Time Markov Chain Model

Markov chains epitomize a class of stochastic process for a wide range of applications. Specifically, discrete time Markov chains (DTMC) is employed to model the transition probabilities between discrete states with the help of the matrices. To examine and forecast the time series the Markov chain model is applied. The most important indicator in macroeconomics is inflation, which persisted in double digits in 1970s and also in last several years. Different states are checked with the model by using inflation rate data form July 2000 to April 2015. A simulation technique used for random sequences of inflation predefine states for one year and take 1st quarter data from it and then model the estimates by maximum likelihood and maximum likelihood with Laplace smoothing methods and check the equilibrium distribution using both techniques. Estimates obtained by Laplace Smoothing technique are reliable because it controls the variation on the Maximum Likelihood estimates.

Shape optimization method for axisymmetric disks based on mesh deformation and smoothing approaches

Axisymmetric disk structures with complex contour curves are widely used in aero-engines. The shape optimization is generally carried out to reduce the stress level of axisymmetric disks. In this article, a shape optimization method for axisymmetric disks based on radial basis function (RBF) mesh deformation and Laplace smoothing approaches is proposed. This method can obtain the optimized reduced control points selection of mesh deformation under the influence of design space based on greedy algorithm. RBF mesh deformation is used to change the axisymmetric contour shape. And after deformation, the local mesh quality is monitored and improved by Laplace smoothing. In this article, two illustrative examples used in aero-engines are carried out to validate the effectiveness of the proposed method, including an independent optimization of a turbine disk and a collaborative optimization of a turbine disk with a deflector for minimizing the maximum equivalent stress. To improve the computational efficiency, a two-dimensional (2D) axisymmetric FE model is established. Compared with initial results, optimized results in two examples obtained by the proposed optimization method reduce the maximum von Mises stress by 8.02% and 9.25%, respectively. It can be concluded that the proposed method has significant potential in the shape optimization design of axisymmetric disks.

Laplacian face blending

AbstractDesigning realistic tridimensional facial models is a challenging task, not only due to the effort and artistic abilities required but also because human visual perception is very tuned to the processing of facial features. For this reason, rather than creating face models from scratch, artists usually start from a scanned model of a real person. In this work, we present a novel method for blending human faces in order to create a new one. In a nutshell, our proposal uses Laplacian smoothing to segregate layers of details from one or more faces, which are then integrated into a base face with the help of an interactive and visual editor. In particular, our method supports blending multiple faces and multiple sub‐regions in those faces. Since our approach is intuitive and relatively easy to implement, it can be integrated into artistic pipelines aiming at designing human face models from preexisting ones.

Laplacian Smoothing Stochastic ADMMs With Differential Privacy Guarantees

Many machine learning tasks such as structured sparse coding and multi-task learning can be converted into an equality constrained optimization problem. The stochastic alternating direction method of multipliers (SADMM) is a popular algorithm to solve such large-scale problems, and has been successfully used in many real-world applications. However, existing SADMMs fail to take into consideration an important issue in their designs, i.e., protecting sensitive information. To address this challenging issue, this paper proposes a novel differential privacy stochastic ADMM framework for solving equality constrained machine learning problems. In particular, to further lift the utility in privacy-preserving equality constrained optimization, a Laplacian smoothing operation is also introduced into our differential privacy ADMM framework, and it can smooth out the Gaussian noise used in the Gaussian mechanism. Then we propose an efficient differentially private variance reduced stochastic ADMM (DP-VRADMM) algorithm with Laplacian smoothing for both strongly convex and general convex objectives. As a by-product, we also present a new differentially private stochastic ADMM algorithm with DP guarantees. In theory, we provide both private guarantees and utility guarantees for the proposed algorithms, which show that Laplacian smoothing can improve the utility bounds of our algorithms. Experimental results on real-world datasets verify our theoretical results and the effectiveness of our algorithms.

4D Structural root architecture modeling from digital twins by X-Ray Computed Tomography

BackgroundBreakthrough imaging technologies may challenge the plant phenotyping bottleneck regarding marker-assisted breeding and genetic mapping. In this context, X-Ray CT (computed tomography) technology can accurately obtain the digital twin of root system architecture (RSA) but computational methods to quantify RSA traits and analyze their changes over time are limited. RSA traits extremely affect agricultural productivity. We develop a spatial–temporal root architectural modeling method based on 4D data from X-ray CT. This novel approach is optimized for high-throughput phenotyping considering the cost-effective time to process the data and the accuracy and robustness of the results. Significant root architectural traits, including root elongation rate, number, length, growth angle, height, diameter, branching map, and volume of axial and lateral roots are extracted from the model based on the digital twin. Our pipeline is divided into two major steps: (i) first, we compute the curve-skeleton based on a constrained Laplacian smoothing algorithm. This skeletal structure determines the registration of the roots over time; (ii) subsequently, the RSA is robustly modeled by a cylindrical fitting to spatially quantify several traits. The experiment was carried out at the Ag Alumni Seed Phenotyping Facility (AAPF) from Purdue University in West Lafayette (IN, USA).ResultsRoots from three samples of tomato plants at two different times and three samples of corn plants at three different times were scanned. Regarding the first step, the PCA analysis of the skeleton is able to accurately and robustly register temporal roots. From the second step, several traits were computed. Two of them were accurately validated using the root digital twin as a ground truth against the cylindrical model: number of branches (RRMSE better than 9%) and volume, reaching a coefficient of determination (R2) of 0.84 and a P < 0.001.ConclusionsThe experimental results support the viability of the developed methodology, being able to provide scalability to a comprehensive analysis in order to perform high throughput root phenotyping.

OpenFOAM for Francis Turbine Transients

The flexibility and fast responsiveness of hydropower systems make them a reliable solution to overcome the intermittency of renewable energy resources and balance the electrical grid. Therefore, investigating the complex flow fields during such operation is essential to increase the reliability and lifetime of future hydropower systems. The current article concerns the utilization of OpenFOAM for the numerical study of Francis turbines during transient load change operations. The details of employed models and numerical schemes are thoroughly explained. The Laplacian smoothing algorithm is applied for the deformation of the guide vane domain. The impact of different mesh diffusivity parameters on both load rejection and acceptance operations is studied. It is shown that general slip boundary conditions cannot be used for slipping points on the guide vane upper and lower surfaces. Instead, different alternatives are introduced and compared. The developed framework is tested on a high-head Francis turbine. Different transient operations are simulated and results are compared to the experimental data. It is shown that OpenFOAM can be employed as a trustworthy CFD solver for numerical investigation of Francis turbines transient operations.

An in-depth numerical analysis of transient flow field in a Francis turbine during shutdown

Power production from intermittent renewable energy resources, such as solar and wind, has increased in the past few decades, leading researchers and engineers to establish techniques to preserve a stable electrical grid. Consequently, hydraulic turbines are being used more frequently in transient operating modes to regulate the grid. The present work provides a comprehensive numerical study on the transient flow field of a high-head Francis turbine model throughout the shutdown sequence. The computations were performed using OpenFOAM, utilizing the SST-SAS turbulence model. A Laplacian smoothing scheme is employed to conduct the mesh deformation of the guide vane domain. The time-averaged draft tube velocity field at the steady Best Efficiency Point (BEP) is validated against experimental data. Then different aspects of the transient flow field in the shutdown sequence are carefully assessed and explained for the first time. Short-Time Fourier Transform (STFT) analysis is carried out on the fluctuating part of the static pressure and force signals. High-amplitude low-frequency oscillations, due to the formation of a Rotating Vortex Rope (RVR) were observed during a specific period of the shutdown sequence. Thereafter, at deep part load conditions, the RVR vanishes and, a wide range of stochastic frequencies are identified at minimum load. A signal coherence analysis was accomplished to distinguish the deterministic and stochastic frequencies. The variation of the velocity field in the draft tube is described in detail with the help of velocity triangles. An in-depth explanation of the formation and variation of vortical structures during the whole sequence is presented. The physical mechanism of formation and destruction of the RVR is thoroughly explained.

ROOT PHENOTYPING FROM X-RAY COMPUTED TOMOGRAPHY: SKELETON EXTRACTION

Abstract. Breakthrough imaging technologies are a potential solution to the plant phenotyping bottleneck in marker-assisted breeding and genetic mapping. X-Ray CT (computed tomography) technology is able to acquire the digital twin of root system architecture (RSA), however, advances in computational methods to digitally model spatial disposition of root system networks are urgently required.We extracted the root skeleton of the digital twin based on 3D data from X-ray CT, which is optimized for high-throughput and robust results. Significant root architectural traits such as number, length, growth angle, elongation rate and branching map can be easily extracted from the skeleton. The curve-skeleton extraction is computed based on a constrained Laplacian smoothing algorithm. This skeletal structure drives the registration procedure in temporal series. The experiment was carried out at the Ag Alumni Seed Phenotyping Facility (AAPF) at Purdue University in West Lafayette (IN, USA). Three samples of tomato root at 2 different times and three samples of corn root at 3 different times were scanned. The skeleton is able to accurately match the shape of the RSA based on a visual inspection.The results based on a visual inspection confirm the feasibility of the proposed methodology, providing scalability to a comprehensive analysis to high throughput root phenotyping.

Designing Parallel Adaptive Laplacian Smoothing for Improving Tetrahedral Mesh Quality on the GPU

Mesh quality is a critical issue in numerical computing because it directly impacts both computational efficiency and accuracy. Tetrahedral meshes are widely used in various engineering and science applications. However, in large-scale and complicated application scenarios, there are a large number of tetrahedrons, and in this case, the improvement of mesh quality is computationally expensive. Laplacian mesh smoothing is a simple mesh optimization method that improves mesh quality by changing the locations of nodes. In this paper, by exploiting the parallelism features of the modern graphics processing unit (GPU), we specifically designed a parallel adaptive Laplacian smoothing algorithm for improving the quality of large-scale tetrahedral meshes. In the proposed adaptive algorithm, we defined the aspect ratio as a metric to judge the mesh quality after each iteration to ensure that every smoothing improves the mesh quality. The adaptive algorithm avoids the shortcoming of the ordinary Laplacian algorithm to create potential invalid elements in the concave area. We conducted 5 groups of comparative experimental tests to evaluate the performance of the proposed parallel algorithm. The results demonstrated that the proposed adaptive algorithm is up to 23 times faster than the serial algorithms; and the accuracy of the tetrahedral mesh is satisfactorily improved after adaptive Laplacian mesh smoothing. Compared with the ordinary Laplacian algorithm, the proposed adaptive Laplacian algorithm is more applicable, and can effectively deal with those tetrahedrons with extremely poor quality. This indicates that the proposed parallel algorithm can be applied to improve the mesh quality in large-scale and complicated application scenarios.