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• A new and efficient nonlinear time domain numerical method for solving electro- and magneto- quasistatic problems. • The governing equations are directly solved in a strong form, but any complicated geometries can be modelled. • The presented method is simple in concept and has a concise derivation process and is easy programming. The transient quasi-static electromagnetic problems with nonlinear conductive effects are under consideration in this paper. The quasi-static electromagnetic problems include electro-quasistatic (EQS) and magneto-quasistatic (MQS) problems, which neglect the wave propagation effects, and are important in the electrical engineering. Numerical computation is an efficient manner for solving these problems, apart from all the existing numerical methods, a new efficient nonlinear time-domain element differential method (TD-EDM) is presented to solve these problems in this paper. In the method, the governing equations and boundary conditions are directly solved in a strong form. The isoparametric elements are used to discrete the geometries, and the first- and second- order derivatives of the shape functions with respect to global coordinates are analytically derived. And the time derivatives of the basic variable are discretized by implicit Euler scheme. In order to handle the nonlinear effects induced by the nonlinear conductivities in the final system of equations, the Newton-Raphson iterative scheme is constructed to solve the nonlinear system of equations at each time step. Finally, several numerical examples are employed to validate the correctness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a simple and efficient numerical method named time domain element differential method (TD-EDM) is proposed to solve electromagnetic wave scattering and radiation problems. In the method, the governing equations as well as boundary conditions are directly solved in a strong-form formulation. The computational domain is discretized using isoparametric elements, and the spatial derivatives of basic variables appearing in the governing equations are calculated by a set of analytical expressions of partial derivatives of the element shape functions with respect to global coordinates, and time derivatives of the basic variable are computed using the Newmark time integration scheme. The absorbing boundary condition (ABC) is applied on the truncation boundary to absorb the outgoing waves, which realizes the numerical modeling of an open infinite region. The incident wave is induced in the total-field region by setting equivalent electromagnetic currents on the total-field/scattering-field (TF/SF) boundary. The TD-EDM equations at the nodes located near the total-field boundary are modified to satisfy the conditions that the electric field values of all nodes involved belong to either the total-field or the scattered field. The near-to-far-field transformation technique is used to calculate the far scattered field, and then the target's Radar Cross Section (RCS) is evaluated. Several numerical results including the radiation from a line current source and the target's Radar Cross Section (RCS) are given to validate correctness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, the element differential method (EDM), a new numerical method proposed recently, is coupled with the multi-domain boundary element method (MDBEM), an improved Boundary Element Method (BEM), for solving general multi-scale heat conduction and elasticity problems. The basic algebraic equations in MDBEM are formulated in terms of displacements/temperatures and surface tractions/heat fluxes, which are the same as those in EDM. Therefore, when coupling these two methods, we don't need to transform the variables such as the equivalent nodal forces into the surface tractions as done in the Finite Element Method (FEM). The key task in the proposed coupled method is to use the displacement/temperature consistency conditions and the surface traction/heat flux equilibrium equations at interface nodes to eliminate all BEM nodes except for those on the interfaces, rather than to iterate. After elimination, the coefficient matrix we get is sparse although a small part is dense. The coupled method inherits the advantages of EDM in flexibility and computational efficiency, and the advantage of BEM in the robustness of treating multi-scale problems. Three numerical examples of general heat conduction and mechanical problems are given to demonstrate the correctness and efficiency of this coupled method. [ABSTRACT FROM AUTHOR]

In this paper, a new type of finite elements, called as Cross-Line Elements (CLEs), are constructed, which have fewest nodes to interpolate physical variables in both two-dimensional (2D) and three-dimensional (3D) problems. These CLEs are then used in a new mesh free method, the Free Element Method (FREM), for solving general 2D and 3D boundary value problems of partial differential equations. FREM is a strong-form element collocation method, combining the advantages of the finite element method and mesh free method in the aspects of setting up shape functions and generating computational meshes through node by node. The distinct feature of FREM is that only one independent element is needed for each collocation node and the element can be freely formed by the nodes surrounding the collocation node. A few numerical examples for 2D and 3D heat conduction and solid mechanics problems will be given to validate the correctness and demonstrate the potential of the constructed elements and proposed numerical method. [ABSTRACT FROM AUTHOR]

In this paper, a novel collocation method is proposed to solve the problems in solid mechanics. In the proposed method, the gradients across the element boundaries are evaluated by a weight-averaging algorithm to achieve higher order continuity. Then, the collocation method is employed to discretize the governing equations for all domain nodes, including the nodes at the interfaces of elements for which the earlier collocation methods have to use the traction-equilibrium equations. As a result of this difference, linear and serendipity elements can be used in the proposed method to reduce the total number of freedoms and easily used for very complex geometries. Moreover, the proposed method significantly improves the accuracy compared with the earlier collocation methods. The provided numerical test cases show that the method can also avoid the degeneration of accuracy when odd-order elements are employed, which further demonstrates the advantages of the proposed method. The proposed method are also leads to better convergency in nonlinear cases and iterative matrix solver. By analyzing the coefficient matrix, we found that the proposed method is better conditioned. The proposed method also yields promising advantages in transient problems for explicit time marching because of the advantage of the collocation methods. • A novel collocation method is proposed for solving problems in solid mechanics. • A weighing procedure is used to evaluate the gradient across the element boundaries. • The proposed method is much accurate and stable for two- and three-dimension problems. • The proposed method can prevent the loss of accuracy when odd-order elements are employed. [ABSTRACT FROM AUTHOR]

The element differential method (EDM) is extended to solve the transient heat transfer problems with phase change. The governing equation of the phase change problem is established in the whole domain by using the effective heat capacity method. Based on the analytical expressions of spatial derivatives of the shape function with respect to the coordinate, discrete equations for internal nodes are generated by employing the governing equation. The interface nodes and the outer boundary nodes meet the flux equilibrium condition. Several numerical examples are given to validate the effectiveness and accuracy of the present method for solving phase change problems. • A new strong-form method (EDM) is extended to deal with phase change heat transfer problems. • EDM can be used to accurately solve phase change problems. • System of equations can be formed without numerical integrals. • A novel collocation method is employed to solve the governing equation of phase change problems. [ABSTRACT FROM AUTHOR]

• Element differential method and gradient-based algorithm are combined to solve inverse phase change problems for the first time. • Three-dimensional (3D) inverse phase change problems are for the first time. • Thermophysical properties and boundary conditions are simultaneously estimated in inverse phase change problems. • The novel multi-parameter identification method is accurate, efficient, robust, and stable. Thermophysical properties and boundary conditions for phase change thermal management systems are challenging to be accurately determined, due to the phase change heat transfer phenomenon and complex working conditions. In this work, the element differential method (EDM) and a gradient-based method are combined to simultaneously predict thermal conductivity, mass specific heat, and boundary heat flux in two-dimensional (2D) and three-dimensional (3D) inverse phase change problems, for the first time. The multi-parameter identification for the 3D physical model with phase change is more general than the previous attempts. Moreover, the effective heat capacity method is employed to deal with phase change problems, to improve efficiency. The sensitivity coefficient is accurately determined by the complex-variable-differentiation method (CVDM) in the multi-parameter prediction. Finally, the effect of measurement points, measurement errors, and initial guessed values on the multi-parameter identification are investigated. This study demonstrates that the present method has good accuracy, efficiency, stability, and robustness in dealing with transient nonlinear inverse problems during phase change process. [ABSTRACT FROM AUTHOR]

The element differential method (EDM) is a robust and efficient strong-form numerical method, which has attracted a lot of attention of the researchers about the numerical methods since proposed. EDM has higher stability than some other strong-form methods and is more efficient than conventional Galerkin FEM. In this paper, the EDM is combined with Solid Isotropic Material with Penalization (SIMP) method to solve 2–3D topological optimization problems under different conditions. The key of the paper is to derive the objective function of second-order EDM element, which is suitable for the SIMP-based topology optimization process. And the sensitivity is solved by Optimality Criteria (OC) method. In the article, topology optimization examples are considered in different mechanics or thermal loads. In the topology optimization of the elastic problem of 3D cantilever beam, the EDM-based SIMP method can reduce the structural compliance by 17∼27% compared with SIMP method based on FEM. And in the heat transfer problems of the 3D plate, the EDM-based SIMP method reduced the compliance by 3.4%. The results show that the present method has good accuracy, efficiency and robustness in topology optimization of 2D and 3D minimum compatibility problems. [ABSTRACT FROM AUTHOR]

In order to accurately identify the temperature-dependent thermal conductivity of solids, a novel method which combines the element differential method (EDM) with the modified Levenberg-Marquardt algorithm (LMA) is firstly proposed to solve the inverse heat conduction problems, where complex variable derivative method (CVDM) is introduced to obtain the derivatives of the observed temperature with respect to the unknown variables. In this paper, the conventional EDM is transformed from the real domain to the complex domain, and then the transient heat conduction problems with complex-variable temperature-dependent thermal conductivity are solved to obtain the numerical temperature solutions of the measuring points using the EDM. The real part of numerical temperature and the corresponding measured temperature are used to build the objective function, and the imaginary part of numerical temperature is used to calculate the sensitivity coefficients of the LMA using the CVDM. Thus, the LMA is modified to stably obtain the sensitivity of the temperature to the unknown variables, and it is used to iteratively optimize the unknown temperature-dependent thermal conductivity by minimizing the objective function. Finally, three numerical examples with the different forms of temperature-dependent thermal conductivity are considered in this paper, and the effects of the different initial values and measurement errors on the inversion results are fully investigated. The results show that the proposed method shows good accuracy, efficiency and robustness in identifying the temperature-dependent thermal conductivity with specific function and that without function form in 2D and 3D models. [ABSTRACT FROM AUTHOR]