Symplectic Eigensolutions Research Articles

A symplectic approach for the fractional heat transfer and thermal damage in 2D biological tissues

PurposeThis paper aims to focus on the accurate analysis of the fractional heat transfer in a two-dimensional (2D) rectangular monolayer tissue with three different kinds of lateral boundary conditions and the quantitative evaluation of the degree of thermal damage and burn depth.Design/methodology/approachA symplectic method is used to analytically solve the fractional heat transfer dual equation in the frequency domain (s-domain). Explicit expressions of the dual vector can be constructed by superposing the symplectic eigensolutions. The solution procedure is rigorously rational without any trial functions. And the accurate predictions of temperature and heat flux in the time domain (t-domain) are derived through numerical inverse Laplace transform.FindingsComparison study shows that the maximum relative error is less than 0.16%, which verifies the accuracy and effectiveness of the proposed method. The results indicate that the model and heat source parameters have a significant effect on temperature and thermal damage. The pulse duration (Δt) of the laser heat source can effectively control the time to reach the peak temperature and the peak slope of the thermal damage curve. The burn depth is closely correlated with exposure temperature and duration. And there exists the delayed effect of fractional order on burn depth.Originality/valueA symplectic approach is presented for the thermal analysis of 2D fractional heat transfer. A unified time-fractional heat transfer model is proposed to describe the anomalous thermal behavior of biological tissue. New findings might provide guidance for temperature prediction and thermal damage assessment of biological tissues during hyperthermia.

Stokes Flow in Cylindrical Containers With Rotating Ends

The Stokes flow in cylindrical containers with rotating ends was studied. Based on the characteristics of the flow, the problem was reduced to the eigenvalue and eigensolution problem of Hamiltonian dual equations with the axial coordinate simulated as the time scale. By means of the completeness of the symplectic eigensolution space and the adjoint symplectic orthogonality relationship between the eigensolutions, the expansion of the solution to the problem was obtained, and the numerical method for calculating the expansion coefficients was given. In the cases of one-end rotating, two-end rotating at the same or opposite angular velocity, the velocity and stress distributions of the flow in the cylindrical containers with different aspect ratios (of the length to the radius), were investigated. The velocity and stress distributions, and the characteristics of the flows under different boundary conditions were revealed.

A Symplectic Analytical Approach for Torsional Buckling of Cylindrical Shells with Asymmetric Local Defects

A torsional buckling model of cylindrical shells with asymmetric local thickness defect is established based on the Hamiltonian system. The critical load and torsional buckling mode of cylindrical shells with defects are obtained by the symplectic eigensolution expansion method, which overcomes the difficulty of constructing the deflection function of the traditional semi-inverse method. Local buckling modes can be captured by this new analytical model with the superposition of symplectic eigensolutions. To ensure accuracy and validity of the symplectic method, the analytical solution with torsional buckling of a cylindrical shell is compared with the classical solution and the finite element method (FEM) solution. The results show that the most detrimental position of the defect is only related to the width of the defect, not to the depth. The local defect changes the circumferential buckling wave number of the cylindrical shell and concentrates the torsional corrugation on the side containing the defect. Torque symmetry is broken due to the asymmetric defect, and the most detrimental defect direction for buckling is the same as the direction of torsional buckling wavelet.

A novel super symplectic analytical singular element for crack propagation along a bimaterial interface

In the last few years, the corresponding author and colleagues proposed a type of symplectic analytical singular elements (SASEs) for various crack problems, such as cracks in plate and shell, bimaterial crack, dynamic crack, crack in viscoelastic media, etc. However, the SASE is still confined to stationary cracks. In this contribution, we propose a new super symplectic analytical singular element (SSASE) for crack propagation along a bimaterial interface. The shape of the SSASE is extended from the normally used circle to general convex polygon, and it can work well on a fixed mesh with triangles or quadrilaterals, thus more proper for modelling crack propagation. The proposed element uses the symplectic eigen solution to define its internal fields, thus most of the advantages of the SASE family are reserved. The proposed element is verified and validated through a few numerical examples. The present contribution has established a new platform for the modelling of the progressive failure process along a bimaterial interface and has also extended the application of the analytical symplectic dual approach to the numerical approach for crack propagations.

Homotopic analysis for post-buckling of cylindrical shells with local thickness defects

A post-buckling model of cylindrical shells with local thickness defects is established based on the Hamiltonian system. The homotopy analysis method and the symplectic eigensolution expansion method are combined to obtain the post-buckling equilibrium path and post-buckling modes of the cylindrical shells with defects. By constructing an eigenvalue buckling problem of the defective cylindrical shell as a zero-order homotopy deformation equation, a significantly better convergence rate can be obtained when compared to the classical perturbation method. The result shows that the defects can induce unique local dents and significantly reduce the ultimate post-buckling bearing capacity of the cylindrical shells. The research finds important and practical applications in many engineering fields that use cylindrical shells as key design components, in particular in aerospace and aeronautical engineering, marine engineering, automobiles, civil engineering, etc.

A Precise Time-Domain Expanding Crack-Tip Element for 3D Planar and Axial Dynamic Cracks

Unlike two-dimensional (2D) crack problems of which the analytical symplectic eigensolutions are available, the analytical symplectic solution of three-dimensional (3D) planar and axial dynamic cracks has not been derived yet. We thus propose a trial displacement field for the 3D planar and axial crack problem using the existing solutions of the 2D crack problems and construct a new crack-tip element on that basis. The recently developed precise integration scheme for 3D planar and axial cracks under dynamic loading (Hu, X. F., Chen, W. H., Zhang, P. and Yao, W. A. [2021] “An explicit crack-tip element for stationary dynamic cracks,” Theor. Appl. Fract. Mech. 112, 102886) is further enhanced. A precise time-domain expanding algorithm is employed for the temporal discretization. All the field variables are expanded using a series expansion scheme, and the dynamic problem is then transformed into a series of quasi-static elastic problems. All the sub-elastic problems are related in a recursive manner. Taking advantage of a smart automatic truncating approach, the number of expanding terms can be determined automatically subjected to a prescribed error tolerance. A few numerical examples are investigated to demonstrate the developed method, numerical solutions indicate that the time dependent stress intensity factors (SIFs) can be calculated in a simple way and the present method has higher efficiency than the recently developed time integration scheme while having comparable accuracy.

A novel isogeometric analysis enriched element for a V-notched one-dimensional hexagonal piezoelectric quasicrystal bi-material

A novel isogeometric analysis enriched element for accurate evaluation of singularities arising at the interface in V-notched one-dimensional hexagonal piezoelectric quasicrystals (PQCs) under anti-plane loading is developed. By taking the advantages of IGA and symplectic methodology, an enriched element centered at the notch tip is constructed by analytical symplectic eigensolutions. Explicit expressions of notch intensity factors and singular multi-physical fields within enriched element are obtained simultaneously. A comparison study between numerical predictions and analytical solutions is performed and excellent agreements are observed. Furthermore, the notch intensity factors for different V-notched PQC bi-material are presented and discussed in detail.

On the symplectic superposition method for free vibration of rectangular thin plates with mixed boundary constraints on an edge

A novel symplectic superposition method has been proposed and developed for plate and shell problems in recent years. The method has yielded many new analytic solutions due to its rigorousness. In this study, the first endeavor is made to further developed the symplectic superposition method for the free vibration of rectangular thin plates with mixed boundary constraints on an edge. The Hamiltonian system-based governing equation is first introduced such that the mathematical techniques in the symplectic space are applied. The solution procedure incorporates separation of variables, symplectic eigen solution and superposition. The analytic solution of an original problem is finally obtained by a set of equations via the equivalence to the superposition of some elaborated subproblems. The natural frequency and mode shape results for representative plates with both clamped and simply supported boundary constraints imposed on the same edge are reported for benchmark use. The present method can be extended to more challenging problems that cannot be solved by conventional analytic methods.

An oblique circular cylinder element for 3D interfacial cracks in composites

This study further extends a developed symplectic analytical singular element (SASE) library to include three-dimensional (3D) cracks in dissimilar materials. We construct a trial symplectic eigen solution for 3D bimaterial interfacial cracks and use it as the crack tip enrichment to define the interior displacement field of a new singular element. The symplectic approach is applied for the numerical modelling of 3D bimaterial cracks for the first time. In the derivation, a non-standard shape of the 3D crack-tip element, i.e., a cylinder with two oblique ends with arbitrary orientations, is considered to better fit the mesh of curved 3D cracks. It is proven that the proposed 3D element has improved the accuracy and is still convenient to use. The present contribution has provided a new insight for choosing effective and efficient crack tip enrichments of 3D interfacial cracks, and is also an important step forward of the SASE library.

Local–global buckling of cylindrical shells with wall thinning defects

The influence of local thinning thickness defects on the buckling of cylindrical shells is investigated. A static buckling model of cylindrical shells with defects is established based on the Hamiltonian system. The complete symplectic eigensolutions of the cylindrical shell are superimposed to derive the buckling modes of the cylindrical shell with defects. A total of seven different cylindrical shells with axisymmetric and non-axisymmetric defects are considered. Local buckling modes can be captured by this new analytical model with symplectic methodology. By constraining the defect volume, the influence of different defect shapes on buckling of the cylindrical shell is investigated. Comparing with step defects and parabolic defects, the analysis concludes that defects of exponential function are more harmful to buckling of cylindrical shells. Highlights This article establishes a cylindrical shell model with arbitrary thickness changes under the Hamiltonian system and evaluates the buckling characteristics of cylindrical shells with thickness defects. A new symplectic approach is developed where the symplectic eigensolutions of the complete cylindrical shell are superimposed to obtain the corresponding buckling modes. Local buckling modes can be captured by this new analytical model. The occurrence of local buckling often drives a significant decrease in critical buckling load. The effects of defect parameters including location, depth, and shape on structure buckling are determined and analyzed.

A symplectic analytical singular element for V-notched analyses in elastic and viscoelastic plane problems

The linear elastic and viscoelastic problems with V-shaped notches under in-plane loading are investigated in this paper. We consider two different sets of boundary conditions on the radial edges: Free-Free case and Clamped-Clamped case. A novel singular finite element (SASE) with arbitrary high-order accuracy for the notched problems is proposed. Firstly, via Laplace transform, the original viscoelastic problem is transformed into a corresponding elastic one. Then we construct the SASE by using elastic symplectic eigen solutions with higher order expanding terms. The SASE can depict the characteristics of displacement fields and singular stress fields in the vicinity of the notch vertex having arbitrary opening angle. By taking advantage of the symplectic eigen solutions in the SASE, Mode I and/or Mode II notch stress intensity factors can be determined directly without any post-processing. Fine finite element meshes are not required. Numerical examples are provided to illustrate the validity and accuracy of the present method.

A new crack-tip singular element for cracks in three-dimensional elastic bodies

Symplectic analytical singular element (SASE) is a special crack-tip element in the framework of finite element method (FEM) for modelling cracks. A group of SASEs, which have been developed by the authors, for various crack problems almost in two-dimensional (2D), but not touched yet more complex ones, i.e., in three-dimensional (3D) domain. The underlying reason lies in the fact that analytical symplectic eigen solution for 3D elastic crack problem is not yet available, a 3D SASE thus cannot be constructed through a simple generalization from the 2D SASEs. This study aims to fill out this 3D gap, we thus use a trail displacement field to construct a 3D SASE. The derived strain and stress fields still possess singularity in the vicinity of crack-tip. Finite element formulation is derived based on the minimum total potential energy principle. It is found that some of key features of a 2D SASE still exist in the developed 3D one, e.g., the SIFs can be calculated accurately without any post-processing. Numerical examples are considered to show the accuracy and performance of the proposed element.

A novel Hamiltonian-based isogeometric analysis of one-dimensional hexagonal piezoelectric quasicrystal with mode III electrically permeable/impermeable cracks

A novel Hamiltonian-based isogeometric analysis is proposed for mode III fracture analysis of piezoelectric quasicrystals (PQCs) with complicated configurations. Two idealized electrical assumptions, namely electrically permeable and impermeable cracks, are considered. The proposed method has two major steps. Firstly, the formulation of isogeometric analysis (IGA) for PQCs is developed and adopted to discretize the overall cracked body. Secondly, analytical solutions near the crack tip are expanded in terms of symplectic eigensolutions in order to change the unknowns of IGA within this region into a set of undetermined coefficients of symplectic series. Therefore, explicit expressions of phonon, phason and electric fields around the crack tip and intensity factors/coefficients are obtained. In numerical examples, several cracked PQCs including a four-leaf-clover-shaped multi-material piezoelectric quasicrystal junction are provided to show the accuracy and effectiveness of the proposed method.

A rational analytical approach for buckling analysis of orthotropic double-nanoplate-systems

A novel analytical Hamiltonian-based approach is proposed for buckling analysis of orthotropic double-nanoplate-systems (DNPSs) under uniaxially compression embedded in an elastic medium. In the Hamiltonian system, the governing equations for in-phase and out-of-phase buckling are established in a unified form based on Eringen’s nonlocal plate theory. The buckling analysis of the orthotropic DNPS is reduced to an eigenproblem in the symplectic space. Analytical buckling equations and buckling mode shape functions can be obtained by the symplectic eigensolutions and boundary conditions simultaneously. Comparison studies demonstrate the accuracy and efficiency of the proposed method. Key influencing factors which may benefit the design of complex 3D mesostructures are studied in detail. Some new results are given also.

A Hamiltonian-based analytical approach for three-dimensional heat conduction of cylinders with specific mixed boundary conditions

A novel analytical symplectic method is introduced to investigate the three-dimensional steady-state heat conduction of cylinders with specific mixed boundary conditions (partial temperature and partial heat flux density). By defining the temperature and heat flux density as the mutually dual variables, the Hamiltonian form of governing equations are established. The original problem is reduced into a symplectic eigenproblem which can be solved by the method of separation of variables and a symplectic sub-system. Exact analytical solution is obtained and expressed in terms of symplectic eigensolutions. Comparison studies demonstrate the accuracy of the proposed method. Some new results are given also.

Interfacial crack analysis between dissimilar viscoelastic media using symplectic analytical singular element

In this paper, a two-dimensional interfacial crack problem between two dissimilar linear viscoelastic media is numerically investigated. The problem can be formulated as a viscoelastic problem. Firstly, the viscoelastic problem in the time domain is transformed into a corresponding elastic one in the Laplace domain using the Laplace transformation. Then a symplectic analytical singular element (SASE) in the framework of finite element method is applied to the obtained elastic problem. The SASE is constructed using bimaterial symplectic eigen solutions with higher order expanding terms. The eigen solutions are derived into explicit forms, and fracture parameters of the problem can be determined directly from expanding coefficients of the eigen solutions. Particularly we investigate how to obtain strain energy release rate, which is an important fracture parameter to viscoelastic fracture problem. Numerical examples are provided to demonstrate the accuracy and effectiveness of the present method.

Accurate fracture analysis of electrically permeable/impermeable cracks in one-dimensional hexagonal piezoelectric quasicrystal junction

An accurate fracture analysis of a multi-material junction of one-dimensional hexagonal quasicrystals with piezoelectric effect is performed by using Hamiltonian mechanics incorporated in the finite element method. Two idealized electrical assumptions, including electrically permeable and impermeable crack-face conditions, are considered. In the Hamiltonian system, the analytical solutions to the multi-material piezoelectric quasicrystal around the crack tip (singular domain) are obtained and expressed in terms of symplectic eigensolutions. Therefore, the large number of nodal unknowns in the singular domain is reduced into a small set of undetermined coefficients of the symplectic series. The unknowns in the non-singular domain remain unchanged. Explicit expressions of phonon stresses, phason stresses, and electric displacement in the singular domain and newly defined fracture parameters are achieved simultaneously. Comparisons are presented to verify the proposed approach and very good agreement is reported. The key influencing parameters of the crack are discussed in detail. The effects of electrical assumptions and positions of the crack on the fracture parameters are discussed in detail.

A Size-Dependent Coupled Symplectic and Finite Element Method for Steady-State Forced Vibration of Built-Up Nanobeam Systems

A novel size-dependent coupled symplectic and finite element method (FEM) is proposed to study the steady-state forced vibration of built-up nanobeam system resting on elastic foundations. The overall system is modeled as a combination of nonlocal Timoshenko beams. A new analytical subsystem modeling with formulation and another numerical subsystem modeling are developed and discussed. In the analytical subsystem model, the uniform nanobeams are modeled and solved by a new approach based on a series of analytical symplectic eigensolutions. The numerical subsystem model applies a nonlocal FEM to solve nonuniform nanobeams. Analytical and numerical solutions are presented, and a proper comparison between the two approaches is established. Comprehensive and accurate numerical result is subsequently presented to illustrate the accuracy and reliability of the coupled method. The new results established are expected to have reference values for future studies.

Viscoelastic crack analysis using symplectic analytical singular element combining with precise time-domain algorithm

In this study, the crack problem in linear viscoelastic material is investigated numerically. The time dependent two-dimensional (2D) viscoelastic crack problem is treated by the precise time-domain expanding algorithm (PTDEA), such that the original problem is transformed into a series of quasi-elastic crack problems. The relationships among these quasi-elastic problems are expressed in terms of the time-domain expanding coefficients of displacement and stress in an improved recursive manner. Then a symplectic analytical singular element (SASE) which has been demonstrated to be effective and efficient for 2D elastic fracture problem is applied to solve the quasi-elastic crack problems obtained above. The SASE is constructed by using the symplectic eigen solutions with higher order expanding terms. An improved convergence criterion employing both displacement and stress for PTDEA is proposed. Taking advantage of the SASE, the stress intensity factors, crack opening and sliding displacements (COD and CSD) and strain energy release rate of the studied problem can be solved directly without any post-processing. Numerical examples show that the results of the present method can be solved accurately and effectively.

A Symplectic Analytical Singular Element for Steady-State Thermal Conduction With Singularities in Anisotropic Material

Modeling of steady-state thermal conduction for crack and v-notch in anisotropic material remains challenging. Conventional numerical methods could bring significant error and the analytical solution should be used to improve the accuracy. In this study, crack and v-notch in anisotropic material are studied. The analytical symplectic eigen solutions are obtained for the first time and used to construct a new symplectic analytical singular element (SASE). The shape functions of the SASE are defined by the obtained eigen solutions (including higher order terms), hence the temperature as well as heat flux fields around the crack/notch tip can be described accurately. The formulation of the stiffness matrix of the SASE is then derived based on a variational principle with two kinds of variables. The nodal variable is transformed into temperature such that the proposed SASE can be connected with conventional finite elements (FE) directly without transition element. Structures of complex geometries and complicated boundary conditions can be analyzed numerically. The generalized flux intensity factors (GFIFs) can be calculated directly without any postprocessing. A few numerical examples are worked out and it is proven that the proposed method is effective for the discussed problem, and the structure can be analyzed accurately and efficiently.