M.T. Alonso Rasgado

A coarse preconditioner for multi-domain boundary element equations

Abstract This paper is concerned with the application of a coarse preconditioner, the generalised minimal residual (GMRES) method and a generalised successive over-relaxation (GSOR) method to linear systems of equations that are derived from boundary integral equations. Attention is restricted to systems of the form ∑ N j =1 H ij x j = c i , i =1,2,…, N , where H ij are matrices, x j and c i are column vectors. The integer N denotes the number of domains and these systems are solved by adapting techniques initially devised for solving single-domain problems. These techniques include parameter matrix accelerated GMRES and GSOR in combination with a multiplicative Schwarz method for non-overlapping domains. The multiplicative Schwarz method is a generalised form of the block Gauss–Seidel method and is called the generalised multi-domain iterative procedure. A new form of coarse grid preconditioning is applied to limit the convergence dependence on block numbers. The coarse preconditioner is obtained from a crude representation of the global system of equations. Attention is restricted to thermal problems with domains connected through resistive thermal barriers. The effect of lowering and increasing the thermal resistance between domains is investigated. The coarse preconditioner requires a more accurate representation on interfaces with lower thermal resistance. Computation times are determined for the iterative procedures and for elimination techniques indicating the relative benefits for problems of this nature.

The 3-D Elastodynamic Boundary Element Method: Semi-Analytical Integration for Linear Isoparametric Triangular Elements

A semi-analytical integration scheme is described in this paper which is designed to reduce the errors incurred when integrals with singular integrands are evaluated numerically. This new scheme can be applied to linear triangular elements for use in steady-state elastodynamic BEM problems and is particularly useful for predicting displacement to high accuracy, close to surfaces for a spectrum of frequencies. The scheme involves the application of Taylor expansions to formulate the integrals into two parts. One part is regular and is evaluated numerically and the other part is singular but sufficiently simple to enable its transforma tion into a line integral. The line integral is solved numerically using Gauss–Legendre quadrature. This approach caters for all the integral types that appear in steady-state elastodynamic boundary elements but, in particular, no special treatment is required for the evaluation of the Cauchy principal value singular integrals. Numerical tests are performed on a simple test-problem for which a known analytical solution exists. The results obtained using the semi-analytical approach are shown to be considerably more accurate than those obtained using standard quadrature methods. Copyright

Experimental Investigation Into the Thermal Behavior of Copper-Alloyed Dies in Pressure Die Casting

The rate of heat extraction during the pressure die casting process is central to both the quality and the cost of finished castings. Recent efforts to reduce the thermal resistance of dies by optimizing the effectiveness of the cooling channels have shown the potential for improvement. Reducing the thermal resistance of the coolant boundary layer means that a significant proportion of the total thermal resistance becomes attributable to the die steel. Further significant reductions in die thermal resistance can be obtained by replacing the steel with copper. This paper investigates the feasibility of using copper dies, reinforced with steel inserts and coated with a thin layer of wear resistant material, which is deposited using the thermal arc spray process. Experimental work relating to the thermal spray process has been undertaken to establish bond strengths and thermal conductivities for various process parameters. Moreover, experimental investigations have been carried out using two copper coated dies, the first of which was a pseudodie block beaten by an infrared heater The second die was tested on a die casting machine and produced zinc alloy castings at a greatly increased production rate when compared to its steel counterpart. The experimental results from the two dies are compared with those predicted by an in-house thermal-cum-stress model based on the boundary element method. Reasonable agreement between the predicted and experimental results is shown and the feasibility of copper-alloyed dies for pressure die casting is established.

An Experimental Investigation on the Effect of Vibration on Casting Surface Finish

One of the failings of the pressure die casting process is that variable surface quality is a common result, This paper provides evidence that die vibration can be used to polish castings as they form during solidification. Evidence is obtained from experiments performed on a purpose built rig that is designed to be representative of the pressure die casting process. The rigs design is such that part of the casting face can be subjected to tangential vibration at prespecified frequencies and amplitudes. The castings made are viewed under a microscope and micrographs produced. These combined with Talysurf readings show that die vibration can be used to generate casting surfaces with a quality significantly superior to that of the die. Supporting evidence is also provided by thermocouple readings and the solution of the inverse parabolic heat equation. The expected increase in the interfacial heat transfer coefficient is obtained with improved casting surface quality. Established in the paper is the effect of vibration frequency and amplitude on the Ra value of the casting surface. It is shown that threshold values for frequency and amplitude exist, below which surface improvement is diminished.

Effects of rolling condition on warm deep drawability of magnesium alloy sheets produced by twin-roll strip casting

Effects of rolling conditions on warm deep drawability of cast magnesium alloy that were hot rolled after roll strip casting were investigated to ascertain the feasibility of twin-roll strip casting process of AZ31B magnesium alloy. Hot rolling and heat treatment conditions were changed to examine which conditions were appropriate for producing AZ31B wrought magnesium alloys after strip casting process. Microscopic observation of the crystals of the manufactured wrought magnesium alloys was performed. It has been found that a limiting drawing ratio of 2.7 was possible in a warm deep drawing test of the cast magnesium alloy sheets after being hot rolled.

Accurate evaluation of integrals present in reciprocity methods

Abstract Reciprocity methods generate boundary integrals of the form ∫ Γ h( x )f(r)g(R) d Γ , where f is singular, r and R are distances measured from a source point and a basis collocation point, respectively. This paper is concerned with the accurate numerical evaluation of integrals of this type. The approach adopted involves the approximation of g(R) by a polynomial p(r), obtained by truncating a Taylor series. The integral ∫ Γ h( x )f(r)g(R) d Γ is equal to ∫ Γ h( x )f(r)(g(R)−p(r)) d Γ+∫ Γ h( x )f(r)p(r) d Γ . The polynomial p(r) is designed to annihilate, where possible, the singularity in ∫ Γ h( x )f(r)(g(R)−p(r)) d Γ and thus facilitate evaluation using standard quadrature. The integral ∫ Γ h( x )f(r)p(r) d Γ is sufficiently simple to be transformed into a contour integral, which can be evaluated numerically using Gauss–Legendre quadrature. To demonstrate implementation of the scheme the thermoelastic BEM is considered. Numerical tests are performed on a simple test-problem for which a known analytical solution exists. The results obtained using the semi-analytical approach are shown to be considerably more accurate than those obtained using standard quadrature methods.

Semi-analytical integration of sub-parametric elements used in the BEM for three-dimensional elastodynamics

Abstract The steady-state elastodynamic boundary element method is an efficient method that can be used in the modelling of vibration systems. The ability to locate natural frequencies and predict displacements close to these frequencies requires the use of high order elements and accurate integration schemes. The possible unboundedness of the displacements at or close to a natural frequency highlights poor numerical conditioning of the algebraic equations resulting from the discretization process, thus making accurate integration schemes a necessity. This paper presents a semi-analytical integration scheme that can be applied to quadratic subparametric triangular elements. The scheme involves subdividing the triangular elements into four triangular subelements. The quadratic shape functions of the original element can then be represented in terms of the linear shape functions of each subelement. A semi-analytical scheme is applied to the integrals involving the linear shape functions of the subelements. Taylor expansions are utilized in the scheme presented to enable the formulation of the integrals into regular and singular parts. Standard numerical schemes are applied to the regular part. The singular part can be transformed into a line integral and evaluated numerically using Gauss–Legendre quadrature. The scheme can handle all integrals appearing in the steady-state elastodynamic BEM with good accuracy. In addition, the Cauchy principal value singular integrals can be dealt with without special treatment. The new scheme is tested by considering integration over two test elements and by application to simple test-problems for which analytical solutions are known.

The effect of vibration on surface finish for semisolid and cast components

Abstract One of the failings of die casting and semisolid forming processes is that variable surface quality is a common result. This can reduce the competitiveness of these processes because secondary finishing operations are required, i.e. machining, polishing, plating, etc. Dies that are producing poor surface finishes cannot at the present time be substituted as it is not known what changes have to be made to give the necessary improvements. The authors have recently established that die vibration can be used to polish surfaces as they form during solidification [J. Manuf. Sci. Eng., Trans. ASME (2001), in press]. Evidence in the form of experiments performed on a purpose built rig has established that tangential vibration at certain frequencies and amplitudes influences surface finish. Micrographs combined with Talysurf readings show that die vibration can be used to generate surfaces with a quality significantly superior to that of the die. This paper is concerned with an investigation into the feasibility of applying die vibration to commercial dies and processes. Boundary element models are utilised that simulate the complex vibrational behaviour of arbitrary shaped dies. Numerical simulation is used to establish the conditions that prevail at the die surface. It is shown that by varying the position and number of sources of vibration, that conditions required for polishing can be approached. Although the research presented is at an early stage, the numerical and experimental results provide evidence that this new technology is potentially applicable to commercial semisolid and casting processes.

Semi‐solid Twin‐roll Casting Process of Magnesium Alloy Sheets

An experimental approach has been performed to ascertain the effectiveness of semi‐solid strip casting using a horizontal twin roll caster. The demand for light‐weight products with high strength has grown recently due to the rapid development of automobile and aircraft technology. One key to such development has been utilization of magnesium alloys, which can potentially reduce the total product weight. However, the problems of utilizing magnesium alloys are still mainly related to high manufacturing cost. One of the solutions to this problem is to develop magnesium casting‐rolling technology in order to produce magnesium sheet products at competitive cost for commercial applications. In this experiment, magnesium alloy AZ31B was used to ascertain the effectiveness of semi‐solid roll strip casting for producing magnesium alloy sheets. The temperature of the molten magnesium, and the roll speeds of the upper and lower rolls, (which could be changed independently), were varied to find an appropriate manufa...

Thermal Modelling In Pressure Die Casting

The pressure die casting process is cyclic and the temperature levels in the die are principally dictated by the total energy received from the casting. It is thus extremely important that any solidification model for the casting is able to predict energy extraction rates to a high degree of accuracy. In this paper an efficient three dimensional hybrid thermal model for the pressure die casting process is described. The finite element method (FEM) is used for modelling heat transfer in the casting, coupled to a boundary element (BE) model for the die. The FEM can efficiently account for the non‐linearity introduced by the release of latent heat on solidification, whereas the BEM is ideally suited for modelling linear heat conduction in the die, as surface temperatures are of principal importance. The FE formulation for the casting is based on a control volume capacitance method, which is shown to provide high accuracy and stability. This method is similar to the apparent and effective heat capacitance met...