Displacement Boundary Conditions Research Articles

Biomechanical evaluation of stability after mandibular sagittal split osteotomy for advancement by Obwegeser–Dal Pont and Puricelli techniques using three-dimensional finite elements

BackgroundThe surgical treatment for mandibular repositioning using a bilateral sagittal split osteotomy (BSSO) favours the development of techniques that result in adequate repair and stability. In Puricelli’s mandibular sagittal split osteotomy (PMSSO) proposal, the vertical lateral cut osteotomy is located in the interradicular space between the lower first molar and second premolar.ObjectivesThis in silico study aimed to investigate the mechanical stability of PMSSO and compare it with the classical Obwegeser–Dal Pont technique for mandibular advancement.Materials and methodsA computational geometric model of the mandible was created in a virtual environment using computer-aided design (CAD) software. After reproducing the advancements, two test groups were developed: GTOD10, Obwegeser–Dal Pont osteotomy, and GTP10, Puricelli osteotomy, both simulating a 10-mm mandibular advancement, allowing for measuring the area of overlap between bone segments. With the geometric changes promoted by the osteotomy, boundary conditions of displacement and force were applied to a CAD software based on finite element analysis (FEA), allowing for quantitative and comparative analysis of the stress and vertical displacement of the mandible, mechanical measurements that may be associated with strength and stiffness.ResultsA 17.48% higher stress was observed in the GTP10 group than in GTOD10. However, the region of highest stress in GTP10 was found in a part of the bone that was still intact and far from the area of fragility caused by lateral vertical osteotomy. In contrast, in GTOD10, the region with high stress was in a less resistant bone region. The GTP10 group showed a 28.73% lower displacement than GTOD10. The area of overlap between the proximal and distal segments of the mandible was 33.13% larger in the GTP10 than in the GTOD10 group.ConclusionThe PMSSO method, performed in large mandibular advancements, keeps the point of highest stress away from the mandibular fragility zone. Considering the same amount of advancement, it also promotes less displacement and larger areas of bone overlap.Clinical relevanceThe results suggest that PMSSO, applied in large mandibular advancement, presents greater postoperative stability.

An improved radial basis reproducing kernel particle method for geometrically nonlinear problem analysis of SMAs

In this paper, the radial basis function (RBF) without shaped parameter is utilized in the radial basis reproducing kernel particle method (RRKPM), and an improved radial basis reproducing kernel particle method (IRRKPM) is proposed. Compared with traditional RKPM, the IRRKPM effectively reduces the impact of different kernel functions on calculation precision, and is further employed to examine geometrically nonlinear problems associated with shape memory alloys (SMAs). The displacement boundary condition is enforced via the penalty function method, while the Galerkin integration method in its weak form, along with the total Lagrangian (TL) approach, is utilized to derive the geometrically nonlinear equations for SMAs within the IRRKPM framework. The equilibrium equations are then solved using the Newton Raphson (N-R) iterative method. The impact of the different penalty factor and the radius control parameter of influence domain on errors is analyzed, the computational precision of the IRRKPM is compared with the RRKPM, and the computational stability is evaluated. Finally, the suitability of the IRRKPM for the analysis of geometrically nonlinearity problems in SMAs are confirmed through specific numerical examples.

Stress solution of static linear elasticity with mixed boundary conditions via adjoint linear operators

We revisit stress problems in linear elasticity to provide a perspective from the geometrical and functional-analytic points of view. For the static stress problem of linear elasticity with mixed boundary conditions we present the associated pair of unbounded adjoint operators. Such a pair is explicitly written for the first time, despite the abundance of the literature on the topic. We use it to find the stress solution as an intersection of the (affinely translated) fundamental subspaces of the adjoint operators. In particular, we treat the equilibrium equation in the operator form, which involves the spaces of traces on a part of the boundary, known as the Lions-Magenes spaces. Our analysis of the pair of adjoint operators for the problem with mixed boundary conditions relies on the properties of the analogous pair of operators for the problem with the displacement boundary conditions, which we also include in the paper.

Physics-Informed Holomorphic Neural Networks (PIHNNs): Solving 2D linear elasticity problems

We propose physics-informed holomorphic neural networks (PIHNNs) as a method to solve boundary value problems where the solution can be represented via holomorphic functions. Specifically, we consider the case of plane linear elasticity and, by leveraging the Kolosov–Muskhelishvili representation of the solution in terms of holomorphic potentials, we train a complex-valued neural network to fulfill stress and displacement boundary conditions while automatically satisfying the governing equations. This is achieved by designing the network to return only approximations that inherently satisfy the Cauchy-Riemann conditions through specific choices of layers and activation functions. To ensure generality, we provide a universal approximation theorem guaranteeing that, under basic assumptions, the proposed holomorphic neural networks can approximate any holomorphic function. Furthermore, we suggest a new tailored weight initialization technique to mitigate the issue of vanishing/exploding gradients. Compared to the standard PINN approach, noteworthy benefits of the proposed method for the linear elasticity problem include a more efficient training, as evaluations are needed solely on the boundary of the domain, lower memory requirements, due to the reduced number of training points, and C∞ regularity of the learned solution. Several benchmark examples are used to verify the correctness of the obtained PIHNN approximations, the substantial benefits over traditional PINNs, and the possibility to deal with non-trivial, multiply-connected geometries via a domain-decomposition strategy.

Stress-based topology optimization using maximum entropy basis functions-based meshless method

AbstractThis paper presents volume-constrained stress minimization-based, topology optimization. The maximum entropy (maxent) basis functions-based meshless method for two-dimensional linear elastic structures is explored. This work focuses to test the effectiveness of the meshless method in handling the stress singularities during the topology optimization process. The commonly used moving least square basis functions are replaced with maximum entropy basis functions, as the latter possess weak Kronecker delta property which leads to the finite element method (FEM) like displacement boundary conditions imposition. The maxent basis functions are calculated once at the beginning of the simulation and then used in optimization at every iteration. Young’s modulus for each background cell is interpolated using the modified solid isotropic material with penalization approach. An open source pre-processor CUBIT is used. A comparison of the proposed approach with the FEM is carried out using a diverse set of problems with simple and complex geometries of structured and unstructured discretization, to establish that maxent-based meshless methods perform better in tackling the stress singularities due to its smooth stress field.

An insight on local defect resonance based on modal decomposition analysis: A two-dimensional case

The resonance behaviors of defects, specifically local defect resonance (LDR), are useful for defect detection in structures. However, there is a lack of intuitive elucidation regarding the underlying mechanisms governing defect resonance. This study aims to analyze the LDR of a horizontal crack in a plate activated by Lamb waves using the modal decomposition method (MDM). The MDM involves an approximate decomposition of the wavefield into finite constituent Lamb wave modes, including propagating, non-propagating, and inhomogeneous modes. The amplitude of each mode is determined according to the continuous boundary conditions of displacement and stress between the damaged zone and the intact zone. Analytical results reveal specific frequencies at which the damaged zone exhibits significantly stronger vibrations than the intact zone, indicating the emergence of LDR. The corresponding wavefields exhibit various resonance patterns, encompassing both in-plane LDR and out-of-plane LDR types. In contrast to finite element simulations and experimental observations, the MDM unveils the dominant constituents of resonance patterns: out-of-plane LDR results from the combination of reflected and transmitted A0 and A1 modes within the damaged zone, while in-plane LDR is formed by the superposition of reflected and transmitted A0, A1, and S0 modes.

Novel multi-degree-of-freedom force–displacement mixed control framework for low loading rate substructure hybrid testing

In substructure hybrid tests, imposing realistic multi-degree-of-freedom (MDOF) displacement and force boundary conditions on a specimen involves controlling several actuators to emulate the MDOF boundary conditions of the experimental substructure. This setup presents significant challenges for decoupling control in MDOF systems. Due to the typically high stiffness of the specimen in the axial direction, force control is preferred over displacement control, which has limited accuracy in this direction. Typically, force control in the axial direction is combined with simultaneous displacement control in the softer direction, forming an MDOF force-displacement mixed control scheme. This scheme transforms force control into displacement control in the inner loop in the axial direction to ensure safe loading. The scheme involves addressing two major challenges: the nonlinear force–displacement transformation and the nonlinear coordinate transformation from the local actuator space (LAS) to the global Cartesian coordinate (GCC) system. To address these challenges, the Modified Newton–Raphson iteration integrated with proportional integral plus feed-forward (MNR-PIFF) and incremental kinematic transformation integrated with proportional integral plus feed-forward (IKT-PIFF) methods are proposed. Based on these methods, a novel MDOF force–displacement mixed control (NMFDC) framework is developed for substructure hybrid testing. A 6-DOF cyclic test simulation using the NMFDC framework was conducted to demonstrate its superiority. Furthermore, a 6-DOF cyclic test and substructure hybrid test at low loading rates were performed to validate the NMFDC framework because of the loading equipment limit. Experimental results and numerical simulations indicate that the NMFDC framework achieves swifter response with lower error levels. For applications in fast and real-time hybrid testing, dynamic loading equipment need to be used, the PIFF controller in the NMFDC framework should be replaced by adaptive time delay compensation in future developments.

Real-Time Simulation of Tube Hydroforming by Integrating Finite-Element Method and Machine Learning

The real-time, full-field simulation of the tube hydroforming process is crucial for deformation monitoring and the timely prediction of defects. However, this is rather difficult for finite-element simulation due to its time-consuming nature. To overcome this drawback, in this paper, a surrogate model framework was proposed by integrating the finite-element method (FEM) and machine learning (ML), in which the basic methodology involved interrupting the computational workflow of the FEM and reassembling it with ML. Specifically, the displacement field, as the primary unknown quantity to be solved using the FEM, was mapped onto the displacement boundary conditions of the tube component with ML. To this end, the titanium tube material as well as the hydroforming process was investigated, and a fairly accurate FEM model was developed based on the CPB06 yield criterion coupled with a simplified Kim–Tuan hardening model. Numerous FEM simulations were performed by varying the loading conditions to generate the training database for ML. Then, a random forest algorithm was applied and trained to develop the surrogate model, in which the grid search method was employed to obtain the optimal combination of the hyperparameters. Sequentially, the principal strain, the effective strain/stress, as well as the wall thickness was derived according to continuum mechanics theories. Although further improvements were required in certain aspects, the developed FEM-ML surrogate model delivered extraordinary accuracy and instantaneity in reproducing multi-physical fields, especially the displacement field and wall-thickness distribution, manifesting its feasibility in the real-time, full-field simulation and monitoring of deformation states.

On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions

We provide (upper and lower) scaling bounds for a singular perturbation model for the cubic-to-tetragonal phase transformation with (partial) displacement boundary data. We illustrate that the order of lamination of the affine displacement data determines the complexity of the microstructure. As in (Rüland and Tribuzio in ESAIM Control Optim. Calc. Var. 29:68, 2023) we heavily exploit careful Fourier space localization methods in distinguishing between the different lamination orders in the data.

Analytic solution for two dimensional beam problems: Pure displacement boundary conditions

The present manuscript delineates the derivation of strong solutions for the linear elasticity problem in a two dimensional rectangular beam. The materials under consideration can exhibit either isotropic or orthotropic properties. Additionally, the analysis is not restricted to slender beams because the ratio between the length and the height of the beam can be arbitrary. The boundary conditions fall into the Dirichlet category, implying that both horizontal and vertical displacements are specified on all four surfaces. The sole requirement is that these surface displacements are continuous functions, although they may not necessarily be smooth. Since the displacements at the surfaces can be arbitrary, there is no need to consider approximations, such as those concerning local (small) boundaries in slender beams. Nonetheless, it is demonstrated that an equivalent principle to the Saint-Venant principle exists for pure displacement boundary conditions. The partial differential equations are solved through the separation of variables method, leading to the identification of two solution types, encompassing both cosine and sine Fourier series. Particular emphasis is placed on evaluating the convergence of these solutions. For two distinct and general examples, it is confirmed that the solutions indeed exist.

A finite-strain plate model for the magneto-mechanical behaviors of hard-magnetic soft material plates

In this paper, we propose a plate model, within the framework of finite-strain elasticity, to study the magneto-mechanical behaviors of hard-magnetic soft material plates. The 2D vector plate equation is derived from the 3D governing equations through a series expansion and truncation approach. The displacement and traction boundary conditions on the edges of plate samples are also proposed. Compared with the other plate or shell theories, the current plate model naturally incorporates the elastic incompressibility and the magneto-mechanical coupling effect of the material, which doesn't involve any a priori hypotheses on the geometry and deformation of the plates. Thus, it is applicable for modeling the large deformations of hard-magnetic soft material plates. To show the efficiency of the plate model, we study a benchmark problem, i.e., the plane-strain bending deformations of plate samples induced by magnetic and mechanical loads. Some analytical solutions of the plate equation system are derived, which provide accurate predictions on the magneto-mechanical response of the plate samples with different thickness-length ratios and magnetization directions. To promote applications of the plate model, we further derive a simplified plate equation system and write out its weak form, which is then implemented into the finite element software COMSOL Multiphysics. The efficiency of the simplified plate equation system is also verified through some typical examples. It is found that the numerical results obtained from the 2D plate model show good consistency with those of the 3D volume model, while the plate model exhibits obvious advantages in the aspects of numerical efficiency.

On the depth of cylindrical indentation of an elastic half-space for different types of displacement boundary conditions

Nonlinear relationships between the penetration depth δ and applied force F for cylindrical indentation of a half-space are derived corresponding to two types of displacement boundary conditions: (i) zero vertical displacement imposed at points x=±b of the free-surface of a half-space, and (ii) zero vertical displacement imposed at a point z=h below the indenter. The requirement that the work done by the indentation force must be equal to the work done by the contact pressure defines the minimum values of b and h for which the indentation is geometrically and physically possible. The minimum value of b is found to be about 10 times greater than the semi-width of the contact zone a=(4FR/πE∗)1/2, with the corresponding minimum value of the indentation depth δmin≈3.5δ0, where δ0=2F/πE∗ is the height of the contact zone under applied force F, and E∗=E/(1−ν2) is the plane strain elastic modulus. The minimum value of h is equal to about 21a in the case ν=1/3, and about 27a in the case ν=1/2. The relationship between h and b, corresponding to the same indentation depth δ≥δmin, is derived and shown to be nearly linear for large values of h/a and b/a. The obtained results are used to estimate the indentation depth of finite-size elastic bodies under different boundary constraints. Comparison with FEM prediction is made.

On modelling strain gradient viscoelasticity of polymer nanocomposites

Theories of generalized continuum mechanics have found great success in the analysis of nanostructures. However, there exists no work on analysing composites whose constituents are generalized continua. The present work fills this gap and studies the strain gradient viscoelasticity of polymeric nanocomposites. The key problem is to assign the nonclassical boundary condition of the representative volume element (RVE). To resolve it, a perturbation field is superposed on the homogeneous displacement boundary condition. The wavelength of perturbation is comparable to the strain gradient characteristic length. Simulations to obtain the macroscopic effective mechanical properties are performed, which agree well with the experimental data. The frequency dependence of the perturbation field is revealed, and it has a clear physical interpretation in terms of the segmental motions of polymer chains.

Influence of Shear Strain on the Deflection of Girders

Numerical calculations are a standard part of modern structural design. Engineers remain particularly interested in real problems where analytical and numerical solutions can be compared with experimental results. Such cases are typical examples of benchmarks because they are used to verify the assumptions introduced. This study shows in detail how shear stresses affect the deflection of a relatively short and high cantilever when the span-to-height ratio of the cross-section is less than five. Such models are frequently used in the design of cantilevers that support heavily loaded beams, for example in the cement industry (e.g., often as structural elements for a heat exchanger system) or for the assessment of short cantilever limit states that appear during excavation in rock sediments. The models are also suitable for designing the various details and joints in the industry of prefabricated elements. This work analyzes in depth the analytical solutions for the displacement field of the linear elastic plane stress theory with two displacement boundary conditions. Also, the solutions were compared with the beam, two-, and three-dimensional numerical models using SAP2000. The results highlight the fundamental principles and solutions behind plane stress and beam theories, with an insight into the advantages and limitations of such models. Doi: 10.28991/CEJ-2024-010-05-04 Full Text: PDF

Efficient atomistic/continuum coupling using lattice Green’s functions

Coupled Atomistic/Continuum (A/C) simulation methods significantly reduce the number of degrees of freedom in the solution of a boundary value problem by using continuum elasticity in domains where defects interact elastically. A/C coupling involves two boundaries: (a) an inner boundary that connects the atomistic and continuum domain and (b) an outer boundary where displacement or force boundary conditions are applied. Here, the lattice Green’s function (LGF)–based flexible boundary condition method is combined with the LGF-based discrete boundary element method to satisfy coupling and boundary conditions at the inner and outer boundaries, respectively, of 3D domains. The coupling algorithms for Dirichlet boundary conditions are presented, with other cases being similar. The LGF is further replaced by a more efficient coarsened LGF method that coarsens the atomistic degrees of freedom at the outer boundary. The scaling of memory and computation versus 3D size and degree of coarsening are discussed. The method and its accuracy are demonstrated using a large 2D problem involving applied forces inside the atomistic domain.

Panel-point model for rigidity and flexibility analysis of rigid origami

In this study, we lay the groundwork for a systematic investigation of the rigidity and flexibility of rigid origami by using the mathematical model referred to as the panel-point model. Rigid origami is commonly known as a type of panel-hinge structure where rigid polygonal panels are connected by rotational hinges, and its motion and stability are often investigated from the perspective of its consistency constraints representing the rigidity and connection conditions of panels. In the proposed methodology, vertex coordinates are directly treated as the variables to represent the rigid origami in the panel-point model, and these variables are constrained by the conditions for the out-of-plane and in-plane rigidity of panels. This model offers several advantages including: 1) the simplicity of polynomial consistency constraints; 2) the ease of incorporating displacement boundary conditions; and 3) the straightforwardness of numerical simulation and visualization. It is anticipated that the presented theories in this article are valuable to a broad audience, including mathematicians, engineers, and architects.

Hybrid vibro-acoustic model reduction for model updating in nuclear power plant pipeline with undetermined boundary conditions

In this work, the hybrid vibro-acoustic model reduction technique that is a physical-modal combined formulation is proposed to accelerate the finite element model updating process of the vibro-acoustic pipeline system. Particularly, the new formulation could provide an effective way of the model updating by preserving the physical DOFs for the direct calibration of the undetermined boundary conditions. The sensitivity based vibro-acoustic model updating is first conducted, and then the undetermined spring constant at the displacement boundary condition is then directly and effectively calibrated by using the proposed hybrid model reduction formulation. The proposed method is implemented in the real nuclear facility to evaluate its performance. In addition, an experimental implementation test using the inverse force identification process is also conducted to demonstrate the reliability of the generated vibro-acoustic FE model through the proposed method.

Micromechanical modeling for longitudinal tensile property of unidirectional CFRP considering dispersion of fiber properties

Longitudinal tensile failure prediction has always been the focus of research on unidirectional (UD) Carbon Fiber Reinforced Polymer (CFRP). In this paper, the longitudinal tensile tests of unidirectional T800 grade carbon fiber reinforced epoxy resin composite were carried out first. The experimental stress–strain curves show that there are obvious progressive damages before the final failures of the specimens. However, this phenomenon is usually neglected in most of the existing investigations and tests to only obtain elastic modulus and ultimate strength. Thus, a Representative Volume Element (RVE) model based on micromechanics was established using finite element method, considering the randomness of fiber spatial distribution and fiber strength. The tensile strength of fiber monofilaments was assumed to follow a statistical two-parameter Weibull distribution. The failure process of UD composite under longitudinal tensile load was simulated by explicit finite element method. The stress–strain curves obtained by simulation are in good agreement with the experimental results. Finally, the influence of different boundary conditions on the calculation of the RVE was studied. It is found that the final failure strengths of the RVE with uniform displacement boundary conditions are similar to the case of periodic boundary conditions, while the former has higher computational efficiency.

Rational generalized mixed finite element method for 2D linear elastic problems

Combining the ideas of rational displacement finite element and generalized mixed finite element, the rational generalized mixed element with 5 nodes is proposed. The finite formulations for the proposed element are derived on the basis of the generalized mixed variational principle, and the two field variables of displacement and stress can be solved simultaneously. Fundamental solutions of planar linear elasticity are adopted as the displacement interpolation function, while polynomials satisfying the equilibrium equation are adopted as the stress interpolation function. Both the displacement and stress boundary conditions can be introduced directly at same time in the proposed method, which contributes to being a more rational model. The solution technique is utilized to analyze a couple of representative static problems, and the results show the properties of excellent accuracy and good convergence. In addition, it is found to be evident for improvement of the accuracy when the stress boundary conditions are applied.

Linear Displacement Boundary Condition Replacing Periodic Boundary Condition to Easily and Efficiently Homogenize the Elastic Properties of Composites

Periodic boundary condition is commonly used in the FE homogenization method for predicting the thermo-mechanical properties of composites. However, its existing numerical algorithms are generally inefficient. Linear displacement boundary condition is much easier and more efficient to implement for the Representative Volume Elements (RVEs) with conformal and nonconformal surface meshes. Aiming at replacing the periodic boundary condition by the linear displacement boundary condition to easily and efficiently predict the thermo-mechanical properties of composites, this paper compares the implementation efficiency and prediction accuracy of the periodic and linear displacement boundary conditions. Regarding the prediction of the elastic properties of composites, the balance of micro-energy and macro-energy of the RVEs imposed the periodic and linear displacement boundary conditions is checked, and the results demonstrate that these two boundary conditions guarantee the Hill lemma (i.e. the balance of micro-energy and macro-energy of the RVEs). Moreover, comparing with the experimental test and the multi-step micro-mechanical method, the effectiveness of the periodic and linear displacement boundary conditions to predict the elastic properties of composites is validated.