聰悟 林 | National Taiwan University (original) (raw)

Papers by 聰悟 林

Earthquake Engineering & Structural Dynamics, May 1, 1990

Half space models which assume the subsoil can be described as an elastic,isotropic, homogeneous ... more Half space models which assume the subsoil can be described as an elastic,isotropic, homogeneous half space are widely used to obtain dynamic compliances and impedance functions. These functions are applicable only in the frequency domain, and thus cannot be used in dynamic analysis of nonlinear structures, which must be performed in the time domain. A convenient method to express the solutions of half space theory in terms of the parameters of a lumped parameter system is presented. These parameters can be used in dynamic analysis of linear or nonlinear structures, considering the effects of soil structure interaction. 906442 Response of piles and pile groups to travelfing SH-waves

International Journal of Statistics and Probability

The Maxwell-Boltzmann speed distribution is the probability distribution that describes the speed... more The Maxwell-Boltzmann speed distribution is the probability distribution that describes the speeds of the particles of ideal gases. The Maxwell-Boltzmann speed distribution is valid for both un-mixed particles (one type of particle) and mixed particles (two types of particles). For mixed particles, both types of particles follow the Maxwell-Boltzmann speed distribution. Also, the most probable speed is inversely proportional to the square root of the mass. This paper proves the Maxwell-Boltzmann speed distribution and the speed ratio of mixed particles using computer-generated data based on Newton’s law of motion. To achieve this, this paper derives the probability density function ψ^ab(u_a;v_a,v_b) of the speed u_a of the particle with mass M_a after the collision of two particles with mass M_a in speed v_a and mass M_b in speed v_b. The function ψ^ab(u_a;v_a,v_b) is obtained through a unique procedure that considers (1) the randomness of the relative direction before a collision...

SYNOPSIS In this investigation, a mathematical hybrid model developed previously is employed to s... more SYNOPSIS In this investigation, a mathematical hybrid model developed previously is employed to study soil-structure interaction of embedded structure. In the analysis, the near field including the embedded structure and its surrounding foundation soil is modelled with a coventional finite element mesh, and the far field is modelled as a semi-infinite medium with a hemi-spherical pit. The impedance functions at the nodes around the special element, which ha~e been determined analytically, can represent the behavior of outgoing propagation of waves. A concept of superposition is proposed to analyze the response of an embedded structure excited by an incoming SH-wave. The governing equations of the whole system will be formulated by enforcing the compatibility and equalibrium conditions at the nodes of the finite mesh. Basing on these equations, the response of the embedded structure and its surrounding ground can be determined accordingly. Numerical results have been obtained, and correlations with available solutions using continuum approaches are studied. The effects of the embedment on the responses are also shown and discussed.

Journal of the Chinese Institute of Engineers, 1992

This paper proposes a simple scheme to decompose an n×n nonpositive definite matrix, A, associate... more This paper proposes a simple scheme to decompose an n×n nonpositive definite matrix, A, associated with simultaneous equations, A X = B, into a triple-factors (lower triangular, diagonal, and upper triangular matrices), i.e., Ã = L D U, without interchanging rows or columns of A, but with A expanded with new rows and new columns to an m×m matrix Ã. Whenever a near-zero diagonal element, say ā ii , is encountered and used as a pivoting element, an appropriate positive real number, say p, is added to this diagonal element, and a new term-px k is also added to the i-th equation, where x k is a new variable called "dummy variable''. If we also add a new equation-px i + px k = 0 to enforce the new added variable x k equal to x i then the modified i-th equation has the same effect as the original equation. Therefore, the original solution X can be found directly from the expanded solution of the modified expanded equation. The method is very useful in solving the following problems: (1) nonlinear problems near the limit state, (2) postbuckling analysis, (3) system equations with constraint conditions, and (4) getting eigenvectors from eigenvalues.

International Journal of Statistics and Probability, 2019

This article derives the probability density function ψξ;x,x'  of the resulting s... more This article derives the probability density function ψξ;x,x'  of the resulting speed ξ  from the collision of two particles with speeds x  and x' . This function had been left unsolved for about 150 years. Then uses two approaches to obtain the Maxwell speed distribution: (1) Numerical iteration: using the equation Pnewξ=0∞0∞ψξ;x,x' ∙Poldx∙Poldx' dxdx'   to get the new speed distribution from the old speed distribution. Also, after 9 iterations, the distribution converges to the Maxwell speed distribution. (2) Analytical integration: using the Maxwell speed distribution as the Poldx , and then getting Pnewξ  from the above integration. The result of Pnewξ  from analytical integration is proved to be exactly the Maxwell speed distribution.

In this work, a periodic method is adopted for a frequency domain analysis of an elevated high-sp... more In this work, a periodic method is adopted for a frequency domain analysis of an elevated high-speed railway bridge. The method assumes the bridge consists of a large number of identical spans which are joined together end-to-end. By using this method, only a one-span model is needed. Several N-span time-domain analysis results using ABAQUS is compared to the periodic analysis result. Our numerical experiments show that the N-span result tends to converge to the periodic analysis result as the N increases.

SUMMARY A three-dimensional hybrid model for the analysis of soilkstructure interaction under dyn... more SUMMARY A three-dimensional hybrid model for the analysis of soilkstructure interaction under dynamic conditions is developed which takes advantage of the desirable features of the finite element and substructure methods and which minimizes their undesirable features. The modelling is achieved by partitioning the total soil-structure system into a near-field and a far-field with a hemispherical interface. The near-field, which consists of the structure to be analysed and a finite region of soil around it, is modelled by finite elements. The semi-infinite far-field is modelled by distributed impedance functions at the interface which are determined by system identification methods. Numerical results indicate that the proposed model makes possible realistic and economical assessment of three-dimensional soil-structure interaction for both surface and embedded structures.

SUMMARY Soil-structure interaction analysis is usually carried out in the frequency domain, becau... more SUMMARY Soil-structure interaction analysis is usually carried out in the frequency domain, because the compliance functions of the half-space are known only in the frequency domain. Since non-linear analysis cannot be carried out in the frequency domain, a system with frequency independent parameters is used to represent the half-space soil medium so that a non-linear analysis in the time domain becomes possible. The objective of this paper is to propose a system with lumped parameters, which are independent of frequency, to represent the half-space soil medium. The proposed frequency independent system consists of a number of real discrete structure elements; thus the existing dynamic analysis programs may be adoptable with little modification. In this paper, the parameters are found by minimizing the sum of the squares of deviations between the steady-state responses of the theoretical half-space model and those of the lumped parameter system over a specified frequency range. Once the parameters have been found, the lumped parameter system can be used in practical applications for time domain dynamic analysis of either linear or non-linear structures. In comparison with the dynamic response of the theoretical half-space model, the lumped parameter system yields satisfactory results.

SUMMARY A new base isolation method is proposed for the protection of structures. Because of the ... more SUMMARY A new base isolation method is proposed for the protection of structures. Because of the efficiency of the isolation devices, the isolated structure may be made to remain elastic throughout major earthquakes. This device consists of two sets of mutually orthogonal free rolling rods under the basement of the structure. Since the coefficient of rolling friction of the rods is very small in practice, the structure can be isolated excellently from the support excitation. In this paper, the analytical method and the response of the isolated system for different parameters, such as the periods of the structure, the coefficient of rolling friction and the masses of rolling rods, are presented. The results indicate that the proposed method is excellent in isolating the structure from support excitations, as expected.

SUMMARY Experimental and analytical studies of base isolation by free rolling rods under basement... more SUMMARY Experimental and analytical studies of base isolation by free rolling rods under basement are described in this paper. The tests of the system, a one-storey, 326-kg structure mounted on a set of free rolling rods, is carried out on a 3" x 3" shaking-table. The dynamic behaviour of the isolated structure is studied and used to verify the analytical results. In the Isolation system, the coefficient of kinetic rolling friction, measured at different angular velocities, ranges from 09007 to OG016. The coefficients are reduced by decreasing the angular velocities. Two earthquakes, a short-period and a long-period motion in Taiwan, are utilized as the input signals. The accelerations experienced by the superstructure are decreased by factors of 56 and 60 in comparison with the fixed-base condition for the two input earthquakes. Also, for each test, the peak relative-to-ground displacement of the basement is nearly equal to the peak ground displacement, and the permanent displacement of the basement is present after the end of the earthquake. Finally, tests of the system with a recentring-force device is undertaken, where a soft spring added to the basement reduces efficiently the permanent displacement. Comparisons show a good agreement between experimental and theoretical results.

A load increment control method called the 'work weighted state vector control method' is propose... more A load increment control method called the 'work weighted state vector control method' is proposed for geometrically nonlinear analysis. The load increment is controlled by keeping the incremental length of the work weighted vector is defined as {{Ajri}, and Ajp are the incremental displacement and incremental load step denoted by Aju' and Ajp'. The length of the state vector is then equal to Z [(A,@ + (Ajp)2] = Z (4~' Ajp')[(Aju /Aju')' + (A,P/A,P')~]. This method can solve snap-through and snap-back problems very efficiently. The effectiveness of the proposed method is demonstrated in the numerical examples. This paper also discusses an algorithm to deal with the limit point and a procedure for automatically determining the direction of the incremental state vector.

Even though electromagnetic wave can be calculated from Maxwell's equations, the cause of elec-tr... more Even though electromagnetic wave can be calculated from Maxwell's equations, the cause of elec-tro-magnetic waves has not been fully understood. This paper proposes a Newton's laws of motion based aether theory to derive identical results as those from Maxwell's equations for free field. The authors suggest that every aether particle has a mass and occupies a volume in space. Every aether particle has translational movement and particle spin movement. The translational movement is similar to the gas particle moving in the air and it does not produce an electromagnetic wave. The particle spin movement generates shear and a spin wave that will be shown to have the same results as Maxwell's equations. Detailed derivation of electromagnetic wave solutions from the proposed aether theory and Maxwell's equations is presented in this paper to show the validation of this model.

Even though the universal gravity force has been formulated and used for more than three hundred ... more Even though the universal gravity force has been formulated and used for more than three hundred years, the cause of the universal gravity force has not been fully understood. This paper proposes a substantial aether model to derive universal gravity force based on Newton's laws of motion and Bernoulli's equation. The authors suggest that every aether particle has a mass and occupies a volume in space. Every aether particle has translational movement and particle spin movement. The particle spin movement does not produce the universal gravity force. The translational movement is similar to the gas particle moving in the air and produces aether pressure. The difference of aether pressure generates an ae-ther flow velocity. The difference of aether flow velocities creates a pressure difference according to Bernoulli's equation. The summation of aether pressure difference around an object is shown to be the universal gravity force. A detailed derivation of universal gravity force from the proposed substantial aether model is presented in this paper to show the validation of this model.

We present an algorithm which uses a new relation vector called the "adjacent elements of each no... more We present an algorithm which uses a new relation vector called the "adjacent elements of each node" and "multiple roots" to replace the "adjacent nodes of each node" and "single root", which are used in the reverse Cuthill-McKee (RCM) method. The "adjacent elements of each node" can be formed easily from the "adjacent nodes of each element" which is a basic given array in a finite element system. The required computer storage is significantly less than the other methods using "adjacent nodes of each node". The listing of FORTRAN subroutines for the proposed algorithm is given. These subroutines can be used directly in the existing finite element system. Since the required storage is small, these subroutines are extremely effective for microcomputers.

This paper proposes a curve for the one-dimensional search method in optimization problems. The s... more This paper proposes a curve for the one-dimensional search method in optimization problems. The search is along a curve instead of a line. This curve is determined by a parameter (u) from the following three points: (1) the initial point X_o , (2) the Cauchy point X_c , (3) the Newton point X_N , and has the following four characteristics: (1) tangents to the steepest descent direction at X_o , (2) passes through X_N , (3) decreases monotonically from X_o to X_N for a quadratic function, (4) no complex computation on the parameter u. Therefore, the proposed method has the following advantages: (1) it is globally convergent, (2) it is locally q-quadratically (or q-superlinearly for quasi-Newton point) convergent, (3) the search procedure is as simple as the line search method.

This paper proposes a penalty function to be used for constrained optimization problems. The prop... more This paper proposes a penalty function to be used for constrained optimization problems. The proposed penalty function is based on two special types of the hyperbolic curve. For the equality constraint, the penalty function is sqrt(x^2 + t^2) , where x= p*g(X), g(X)=0 is the constraint, t is a shape parameter, p is a scale factor, and X contains the design variables. For the inequality constraint, the penalty function is sqrt(x^2 + t^2) – x, where x = p*g(X), and p > 0 is the constraint for g(X) >= 0, or p < 0 for g(X) <= 0. These penalty functions have the advantages of being defined everywhere, accurate and differentiable. Two extended penalty functions are also proposed to dominate the infinite negative objective function in some cases. One is of the power type, another is of the exponential type. These two extended penalty functions are continuous up to their second derivatives. Therefore, they can be used in almost all of the optimization methods.

A procedure for implementing constraint relations among finite element nodal degrees of freedom i... more A procedure for implementing constraint relations among finite element nodal degrees of freedom is outlined. Rather than imposing the constraint relations on the global stiffness or mass matrix as the conventional approach, this procedure is based on the element formulation level in that the element matrices and vectors are properly converted to account for the effect of constraint relations before the global assemblage phase. Adjustment of the global matrix profile for constraint relations and matrix pivoting in equation solving are thus avoided , and the approach can be easily incorporated into existing program systems. In this paper, the theoretical treatment for the proposed procedure is given, and the software implementation aspects are also discussed.

This paper proposes an adaptive modification method to transform the objective function with a st... more This paper proposes an adaptive modification method to transform the objective function with a stationary point to an objective function with a minima point, such that search methods can be used to find the stationary point. The stationary point can be a saddle point in addition to a minima or a maxima. Therefore, this method can be used to transform a constrained optimization by applying Lagrange multipliers to an unconstrained optimization problem. A quadratic term, ½<X - X_N>[D]{X-X_N}, is added to the original function such that the modified function is a minima at the Newton point {X_N} of the original function, where [D] is a diagonal matrix to make the modified Hessian matrix [H_O}+[D] positive definite, and [H_O] is the original Hessian matrix at the initial point {X_O} .

This paper proposes a simple scheme to decompose an n×n nonpositive definite matrix, A, associate... more This paper proposes a simple scheme to decompose an n×n nonpositive definite matrix, A, associated with simultaneous equations, A X = B, into a triple-factors (lower triangular, diagonal, and upper triangular matrices), i.e., Ã = L D U, without interchanging rows or columns of A, but with A expanded with new rows and new columns to an m×m matrix Ã. Whenever a near-zero diagonal element, say ā ii , is encountered and used as a pivoting element, an appropriate positive real number, say p, is added to this diagonal element, and a new term —px k is also added to the i-th equation, where x k is a new variable called "dummy variable''. If we also add a new equation —px i + px k = 0 to enforce the new added variable x k equal to x i then the modified i-th equation has the same effect as the original equation. Therefore, the original solution X can be found directly from the expanded solution of the modified expanded equation. The method is very useful in solving the following problems: (1) nonlinear problems near the limit state, (2) postbuckl-ing analysis, (3) system equations with constraint conditions, and (4) getting eigenvectors from eigenvalues. AX=B,

Conference Presentations by 聰悟 林

In this investigation, a mathematical hybrid model developed previously is employed to study soil... more In this investigation, a mathematical hybrid model developed previously is employed to study soil-structure interaction of embedded structure. In the analysis, the near field including the embedded structure and its surrounding foundation soil is modelled with a conventional finite element mesh, and the far field is modelled as a semi-infinite medium with a hemi-spherical pit. The impedance functions at the nodes around the special element, which have been determinated analytically, can represent the behavior of outgoing propagation of waves. A concept of superposition is proposed to analyze the response of an embedded structure excited by an incoming SH-wave. The governing equations of the whole system will be formulated by enforcing the compatibility and equalibrium conditions of the nodes of the finite mesh. Basing on these equations, the response of the embedded structure and its surrounding ground can be determined accordingly. Numerical results have been obtained, and the correlations with available solutions using continuum approaches are atudied. The effects of the embedment on the responses are also shown and discussed.

Earthquake Engineering & Structural Dynamics, May 1, 1990

Half space models which assume the subsoil can be described as an elastic,isotropic, homogeneous ... more Half space models which assume the subsoil can be described as an elastic,isotropic, homogeneous half space are widely used to obtain dynamic compliances and impedance functions. These functions are applicable only in the frequency domain, and thus cannot be used in dynamic analysis of nonlinear structures, which must be performed in the time domain. A convenient method to express the solutions of half space theory in terms of the parameters of a lumped parameter system is presented. These parameters can be used in dynamic analysis of linear or nonlinear structures, considering the effects of soil structure interaction. 906442 Response of piles and pile groups to travelfing SH-waves

International Journal of Statistics and Probability

The Maxwell-Boltzmann speed distribution is the probability distribution that describes the speed... more The Maxwell-Boltzmann speed distribution is the probability distribution that describes the speeds of the particles of ideal gases. The Maxwell-Boltzmann speed distribution is valid for both un-mixed particles (one type of particle) and mixed particles (two types of particles). For mixed particles, both types of particles follow the Maxwell-Boltzmann speed distribution. Also, the most probable speed is inversely proportional to the square root of the mass. This paper proves the Maxwell-Boltzmann speed distribution and the speed ratio of mixed particles using computer-generated data based on Newton’s law of motion. To achieve this, this paper derives the probability density function ψ^ab(u_a;v_a,v_b) of the speed u_a of the particle with mass M_a after the collision of two particles with mass M_a in speed v_a and mass M_b in speed v_b. The function ψ^ab(u_a;v_a,v_b) is obtained through a unique procedure that considers (1) the randomness of the relative direction before a collision...

SYNOPSIS In this investigation, a mathematical hybrid model developed previously is employed to s... more SYNOPSIS In this investigation, a mathematical hybrid model developed previously is employed to study soil-structure interaction of embedded structure. In the analysis, the near field including the embedded structure and its surrounding foundation soil is modelled with a coventional finite element mesh, and the far field is modelled as a semi-infinite medium with a hemi-spherical pit. The impedance functions at the nodes around the special element, which ha~e been determined analytically, can represent the behavior of outgoing propagation of waves. A concept of superposition is proposed to analyze the response of an embedded structure excited by an incoming SH-wave. The governing equations of the whole system will be formulated by enforcing the compatibility and equalibrium conditions at the nodes of the finite mesh. Basing on these equations, the response of the embedded structure and its surrounding ground can be determined accordingly. Numerical results have been obtained, and correlations with available solutions using continuum approaches are studied. The effects of the embedment on the responses are also shown and discussed.

Journal of the Chinese Institute of Engineers, 1992

This paper proposes a simple scheme to decompose an n×n nonpositive definite matrix, A, associate... more This paper proposes a simple scheme to decompose an n×n nonpositive definite matrix, A, associated with simultaneous equations, A X = B, into a triple-factors (lower triangular, diagonal, and upper triangular matrices), i.e., Ã = L D U, without interchanging rows or columns of A, but with A expanded with new rows and new columns to an m×m matrix Ã. Whenever a near-zero diagonal element, say ā ii , is encountered and used as a pivoting element, an appropriate positive real number, say p, is added to this diagonal element, and a new term-px k is also added to the i-th equation, where x k is a new variable called "dummy variable''. If we also add a new equation-px i + px k = 0 to enforce the new added variable x k equal to x i then the modified i-th equation has the same effect as the original equation. Therefore, the original solution X can be found directly from the expanded solution of the modified expanded equation. The method is very useful in solving the following problems: (1) nonlinear problems near the limit state, (2) postbuckling analysis, (3) system equations with constraint conditions, and (4) getting eigenvectors from eigenvalues.

International Journal of Statistics and Probability, 2019

This article derives the probability density function ψξ;x,x'  of the resulting s... more This article derives the probability density function ψξ;x,x'  of the resulting speed ξ  from the collision of two particles with speeds x  and x' . This function had been left unsolved for about 150 years. Then uses two approaches to obtain the Maxwell speed distribution: (1) Numerical iteration: using the equation Pnewξ=0∞0∞ψξ;x,x' ∙Poldx∙Poldx' dxdx'   to get the new speed distribution from the old speed distribution. Also, after 9 iterations, the distribution converges to the Maxwell speed distribution. (2) Analytical integration: using the Maxwell speed distribution as the Poldx , and then getting Pnewξ  from the above integration. The result of Pnewξ  from analytical integration is proved to be exactly the Maxwell speed distribution.

In this work, a periodic method is adopted for a frequency domain analysis of an elevated high-sp... more In this work, a periodic method is adopted for a frequency domain analysis of an elevated high-speed railway bridge. The method assumes the bridge consists of a large number of identical spans which are joined together end-to-end. By using this method, only a one-span model is needed. Several N-span time-domain analysis results using ABAQUS is compared to the periodic analysis result. Our numerical experiments show that the N-span result tends to converge to the periodic analysis result as the N increases.

SUMMARY A three-dimensional hybrid model for the analysis of soilkstructure interaction under dyn... more SUMMARY A three-dimensional hybrid model for the analysis of soilkstructure interaction under dynamic conditions is developed which takes advantage of the desirable features of the finite element and substructure methods and which minimizes their undesirable features. The modelling is achieved by partitioning the total soil-structure system into a near-field and a far-field with a hemispherical interface. The near-field, which consists of the structure to be analysed and a finite region of soil around it, is modelled by finite elements. The semi-infinite far-field is modelled by distributed impedance functions at the interface which are determined by system identification methods. Numerical results indicate that the proposed model makes possible realistic and economical assessment of three-dimensional soil-structure interaction for both surface and embedded structures.

SUMMARY Soil-structure interaction analysis is usually carried out in the frequency domain, becau... more SUMMARY Soil-structure interaction analysis is usually carried out in the frequency domain, because the compliance functions of the half-space are known only in the frequency domain. Since non-linear analysis cannot be carried out in the frequency domain, a system with frequency independent parameters is used to represent the half-space soil medium so that a non-linear analysis in the time domain becomes possible. The objective of this paper is to propose a system with lumped parameters, which are independent of frequency, to represent the half-space soil medium. The proposed frequency independent system consists of a number of real discrete structure elements; thus the existing dynamic analysis programs may be adoptable with little modification. In this paper, the parameters are found by minimizing the sum of the squares of deviations between the steady-state responses of the theoretical half-space model and those of the lumped parameter system over a specified frequency range. Once the parameters have been found, the lumped parameter system can be used in practical applications for time domain dynamic analysis of either linear or non-linear structures. In comparison with the dynamic response of the theoretical half-space model, the lumped parameter system yields satisfactory results.

SUMMARY A new base isolation method is proposed for the protection of structures. Because of the ... more SUMMARY A new base isolation method is proposed for the protection of structures. Because of the efficiency of the isolation devices, the isolated structure may be made to remain elastic throughout major earthquakes. This device consists of two sets of mutually orthogonal free rolling rods under the basement of the structure. Since the coefficient of rolling friction of the rods is very small in practice, the structure can be isolated excellently from the support excitation. In this paper, the analytical method and the response of the isolated system for different parameters, such as the periods of the structure, the coefficient of rolling friction and the masses of rolling rods, are presented. The results indicate that the proposed method is excellent in isolating the structure from support excitations, as expected.

SUMMARY Experimental and analytical studies of base isolation by free rolling rods under basement... more SUMMARY Experimental and analytical studies of base isolation by free rolling rods under basement are described in this paper. The tests of the system, a one-storey, 326-kg structure mounted on a set of free rolling rods, is carried out on a 3" x 3" shaking-table. The dynamic behaviour of the isolated structure is studied and used to verify the analytical results. In the Isolation system, the coefficient of kinetic rolling friction, measured at different angular velocities, ranges from 09007 to OG016. The coefficients are reduced by decreasing the angular velocities. Two earthquakes, a short-period and a long-period motion in Taiwan, are utilized as the input signals. The accelerations experienced by the superstructure are decreased by factors of 56 and 60 in comparison with the fixed-base condition for the two input earthquakes. Also, for each test, the peak relative-to-ground displacement of the basement is nearly equal to the peak ground displacement, and the permanent displacement of the basement is present after the end of the earthquake. Finally, tests of the system with a recentring-force device is undertaken, where a soft spring added to the basement reduces efficiently the permanent displacement. Comparisons show a good agreement between experimental and theoretical results.

A load increment control method called the 'work weighted state vector control method' is propose... more A load increment control method called the 'work weighted state vector control method' is proposed for geometrically nonlinear analysis. The load increment is controlled by keeping the incremental length of the work weighted vector is defined as {{Ajri}, and Ajp are the incremental displacement and incremental load step denoted by Aju' and Ajp'. The length of the state vector is then equal to Z [(A,@ + (Ajp)2] = Z (4~' Ajp')[(Aju /Aju')' + (A,P/A,P')~]. This method can solve snap-through and snap-back problems very efficiently. The effectiveness of the proposed method is demonstrated in the numerical examples. This paper also discusses an algorithm to deal with the limit point and a procedure for automatically determining the direction of the incremental state vector.

Even though electromagnetic wave can be calculated from Maxwell's equations, the cause of elec-tr... more Even though electromagnetic wave can be calculated from Maxwell's equations, the cause of elec-tro-magnetic waves has not been fully understood. This paper proposes a Newton's laws of motion based aether theory to derive identical results as those from Maxwell's equations for free field. The authors suggest that every aether particle has a mass and occupies a volume in space. Every aether particle has translational movement and particle spin movement. The translational movement is similar to the gas particle moving in the air and it does not produce an electromagnetic wave. The particle spin movement generates shear and a spin wave that will be shown to have the same results as Maxwell's equations. Detailed derivation of electromagnetic wave solutions from the proposed aether theory and Maxwell's equations is presented in this paper to show the validation of this model.

Even though the universal gravity force has been formulated and used for more than three hundred ... more Even though the universal gravity force has been formulated and used for more than three hundred years, the cause of the universal gravity force has not been fully understood. This paper proposes a substantial aether model to derive universal gravity force based on Newton's laws of motion and Bernoulli's equation. The authors suggest that every aether particle has a mass and occupies a volume in space. Every aether particle has translational movement and particle spin movement. The particle spin movement does not produce the universal gravity force. The translational movement is similar to the gas particle moving in the air and produces aether pressure. The difference of aether pressure generates an ae-ther flow velocity. The difference of aether flow velocities creates a pressure difference according to Bernoulli's equation. The summation of aether pressure difference around an object is shown to be the universal gravity force. A detailed derivation of universal gravity force from the proposed substantial aether model is presented in this paper to show the validation of this model.

We present an algorithm which uses a new relation vector called the "adjacent elements of each no... more We present an algorithm which uses a new relation vector called the "adjacent elements of each node" and "multiple roots" to replace the "adjacent nodes of each node" and "single root", which are used in the reverse Cuthill-McKee (RCM) method. The "adjacent elements of each node" can be formed easily from the "adjacent nodes of each element" which is a basic given array in a finite element system. The required computer storage is significantly less than the other methods using "adjacent nodes of each node". The listing of FORTRAN subroutines for the proposed algorithm is given. These subroutines can be used directly in the existing finite element system. Since the required storage is small, these subroutines are extremely effective for microcomputers.

This paper proposes a curve for the one-dimensional search method in optimization problems. The s... more This paper proposes a curve for the one-dimensional search method in optimization problems. The search is along a curve instead of a line. This curve is determined by a parameter (u) from the following three points: (1) the initial point X_o , (2) the Cauchy point X_c , (3) the Newton point X_N , and has the following four characteristics: (1) tangents to the steepest descent direction at X_o , (2) passes through X_N , (3) decreases monotonically from X_o to X_N for a quadratic function, (4) no complex computation on the parameter u. Therefore, the proposed method has the following advantages: (1) it is globally convergent, (2) it is locally q-quadratically (or q-superlinearly for quasi-Newton point) convergent, (3) the search procedure is as simple as the line search method.

This paper proposes a penalty function to be used for constrained optimization problems. The prop... more This paper proposes a penalty function to be used for constrained optimization problems. The proposed penalty function is based on two special types of the hyperbolic curve. For the equality constraint, the penalty function is sqrt(x^2 + t^2) , where x= p*g(X), g(X)=0 is the constraint, t is a shape parameter, p is a scale factor, and X contains the design variables. For the inequality constraint, the penalty function is sqrt(x^2 + t^2) – x, where x = p*g(X), and p > 0 is the constraint for g(X) >= 0, or p < 0 for g(X) <= 0. These penalty functions have the advantages of being defined everywhere, accurate and differentiable. Two extended penalty functions are also proposed to dominate the infinite negative objective function in some cases. One is of the power type, another is of the exponential type. These two extended penalty functions are continuous up to their second derivatives. Therefore, they can be used in almost all of the optimization methods.

A procedure for implementing constraint relations among finite element nodal degrees of freedom i... more A procedure for implementing constraint relations among finite element nodal degrees of freedom is outlined. Rather than imposing the constraint relations on the global stiffness or mass matrix as the conventional approach, this procedure is based on the element formulation level in that the element matrices and vectors are properly converted to account for the effect of constraint relations before the global assemblage phase. Adjustment of the global matrix profile for constraint relations and matrix pivoting in equation solving are thus avoided , and the approach can be easily incorporated into existing program systems. In this paper, the theoretical treatment for the proposed procedure is given, and the software implementation aspects are also discussed.

This paper proposes an adaptive modification method to transform the objective function with a st... more This paper proposes an adaptive modification method to transform the objective function with a stationary point to an objective function with a minima point, such that search methods can be used to find the stationary point. The stationary point can be a saddle point in addition to a minima or a maxima. Therefore, this method can be used to transform a constrained optimization by applying Lagrange multipliers to an unconstrained optimization problem. A quadratic term, ½<X - X_N>[D]{X-X_N}, is added to the original function such that the modified function is a minima at the Newton point {X_N} of the original function, where [D] is a diagonal matrix to make the modified Hessian matrix [H_O}+[D] positive definite, and [H_O] is the original Hessian matrix at the initial point {X_O} .

This paper proposes a simple scheme to decompose an n×n nonpositive definite matrix, A, associate... more This paper proposes a simple scheme to decompose an n×n nonpositive definite matrix, A, associated with simultaneous equations, A X = B, into a triple-factors (lower triangular, diagonal, and upper triangular matrices), i.e., Ã = L D U, without interchanging rows or columns of A, but with A expanded with new rows and new columns to an m×m matrix Ã. Whenever a near-zero diagonal element, say ā ii , is encountered and used as a pivoting element, an appropriate positive real number, say p, is added to this diagonal element, and a new term —px k is also added to the i-th equation, where x k is a new variable called "dummy variable''. If we also add a new equation —px i + px k = 0 to enforce the new added variable x k equal to x i then the modified i-th equation has the same effect as the original equation. Therefore, the original solution X can be found directly from the expanded solution of the modified expanded equation. The method is very useful in solving the following problems: (1) nonlinear problems near the limit state, (2) postbuckl-ing analysis, (3) system equations with constraint conditions, and (4) getting eigenvectors from eigenvalues. AX=B,

In this investigation, a mathematical hybrid model developed previously is employed to study soil... more In this investigation, a mathematical hybrid model developed previously is employed to study soil-structure interaction of embedded structure. In the analysis, the near field including the embedded structure and its surrounding foundation soil is modelled with a conventional finite element mesh, and the far field is modelled as a semi-infinite medium with a hemi-spherical pit. The impedance functions at the nodes around the special element, which have been determinated analytically, can represent the behavior of outgoing propagation of waves. A concept of superposition is proposed to analyze the response of an embedded structure excited by an incoming SH-wave. The governing equations of the whole system will be formulated by enforcing the compatibility and equalibrium conditions of the nodes of the finite mesh. Basing on these equations, the response of the embedded structure and its surrounding ground can be determined accordingly. Numerical results have been obtained, and the correlations with available solutions using continuum approaches are atudied. The effects of the embedment on the responses are also shown and discussed.

This study presents the closed expansion form of Duhamel's integral and the general expressions f... more This study presents the closed expansion form of Duhamel's integral and the general expressions for the numerical evaluation of the response of a single degree of freedom system. The external dynamic loading are expressed in polynomial form. A subroutine which uses an equal time spaced input record and a subroutine which can accept an unequal time spaced input record were developed.
These subroutines were tested with sinusoidal loading data and the numerical results obtained were compared with the exact solution.